Introduction and Overview
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The Laplace Transform
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The Inverse Laplace Transform
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- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31

Solving ODEs with the Laplace Transform
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[{"Name":"Introduction and Overview","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction and Overview","Duration":"11m 33s","ChapterTopicVideoID":7842,"CourseChapterTopicPlaylistID":4245,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.910","Text":"This clip is just an introduction to something called the Laplace transform,"},{"Start":"00:05.910 ","End":"00:08.610","Text":"which is a concept in mathematics in itself,"},{"Start":"00:08.610 ","End":"00:11.070","Text":"but it also has uses for"},{"Start":"00:11.070 ","End":"00:13.920","Text":"ordinary differential equations and that\u0027s why we\u0027re learning"},{"Start":"00:13.920 ","End":"00:16.845","Text":"it and it also has something called an inverse."},{"Start":"00:16.845 ","End":"00:18.870","Text":"Let\u0027s start with the definition."},{"Start":"00:18.870 ","End":"00:24.580","Text":"Suppose we have some function of g(t) and I want to mention that t"},{"Start":"00:24.580 ","End":"00:30.410","Text":"is the usual variable we use when we take Laplace transforms as opposed to x,"},{"Start":"00:30.410 ","End":"00:32.450","Text":"it\u0027s customary to use t,"},{"Start":"00:32.450 ","End":"00:34.775","Text":"so the Laplace transform of a function,"},{"Start":"00:34.775 ","End":"00:37.220","Text":"and it\u0027s denoted like this."},{"Start":"00:37.220 ","End":"00:43.280","Text":"This is some a curly L. What it is is it\u0027s a new function,"},{"Start":"00:43.280 ","End":"00:46.310","Text":"but in a different variable, s,"},{"Start":"00:46.310 ","End":"00:52.160","Text":"usually G(s) and the convention is that if we transform g,"},{"Start":"00:52.160 ","End":"00:56.075","Text":"we get G and f would get F and so on."},{"Start":"00:56.075 ","End":"00:59.465","Text":"We define as follows."},{"Start":"00:59.465 ","End":"01:05.020","Text":"G(s) which is the Laplace transform of g(t),"},{"Start":"01:05.020 ","End":"01:09.055","Text":"is defined as an integral from 0 to infinity,"},{"Start":"01:09.055 ","End":"01:12.850","Text":"e^minus st, g(t) dt."},{"Start":"01:12.850 ","End":"01:16.450","Text":"The integral of e^minus st are always there."},{"Start":"01:16.450 ","End":"01:19.450","Text":"What changes is the function I put here,"},{"Start":"01:19.450 ","End":"01:22.750","Text":"and that\u0027s our particular function, g(t)."},{"Start":"01:22.750 ","End":"01:26.065","Text":"Now notice that after we take the integral,"},{"Start":"01:26.065 ","End":"01:30.130","Text":"we don\u0027t get t anymore because the integration is respect to t,"},{"Start":"01:30.130 ","End":"01:35.500","Text":"what we\u0027re left with is a function of s. Also note that this is an improper integral,"},{"Start":"01:35.500 ","End":"01:42.920","Text":"so sometimes we might have to put some number here and let it tend to infinity."},{"Start":"01:42.920 ","End":"01:45.320","Text":"We\u0027ll take the limit as it goes to infinity."},{"Start":"01:45.320 ","End":"01:47.570","Text":"Now this is all very abstract and theoretical,"},{"Start":"01:47.570 ","End":"01:51.500","Text":"so we\u0027re going to need some examples though, bring 2 examples."},{"Start":"01:51.500 ","End":"01:54.350","Text":"The first example we\u0027re going to show you that if we take"},{"Start":"01:54.350 ","End":"01:58.505","Text":"the constant function g(t) is the constant 1,"},{"Start":"01:58.505 ","End":"02:00.410","Text":"then its Laplace transform,"},{"Start":"02:00.410 ","End":"02:06.160","Text":"G(s) is 1 over s and I want to demonstrate this for you,"},{"Start":"02:06.160 ","End":"02:11.965","Text":"so G(s) which is the transform of the number 1, the function 1."},{"Start":"02:11.965 ","End":"02:14.480","Text":"Using the definition and in our case,"},{"Start":"02:14.480 ","End":"02:16.555","Text":"g(t) is 1,"},{"Start":"02:16.555 ","End":"02:22.295","Text":"so we put the 1 inside the definition from here and this is the integral we get."},{"Start":"02:22.295 ","End":"02:25.100","Text":"Turns out this integral which is an improper integral,"},{"Start":"02:25.100 ","End":"02:27.965","Text":"converges when s is positive."},{"Start":"02:27.965 ","End":"02:29.570","Text":"If s is positive,"},{"Start":"02:29.570 ","End":"02:31.340","Text":"and you\u0027ll see why in a moment,"},{"Start":"02:31.340 ","End":"02:32.645","Text":"why I say positive."},{"Start":"02:32.645 ","End":"02:38.240","Text":"First of all, the integral of this is e^minus st over the coefficient of t,"},{"Start":"02:38.240 ","End":"02:39.545","Text":"which is minus s,"},{"Start":"02:39.545 ","End":"02:41.605","Text":"s is a constant here."},{"Start":"02:41.605 ","End":"02:43.775","Text":"Now when s is positive,"},{"Start":"02:43.775 ","End":"02:48.139","Text":"then we get here e^minus infinity."},{"Start":"02:48.139 ","End":"02:52.520","Text":"When we let t go to infinity and e^minus infinity is 0,"},{"Start":"02:52.520 ","End":"02:54.515","Text":"so the first part is 0."},{"Start":"02:54.515 ","End":"02:57.140","Text":"For the upper limit, the lower limit is 0,"},{"Start":"02:57.140 ","End":"03:02.930","Text":"e^0 is 1 and so we get minus 1 over minus s and the minus,"},{"Start":"03:02.930 ","End":"03:06.370","Text":"with the minus will give us 1 over s,"},{"Start":"03:06.370 ","End":"03:08.285","Text":"s has to be bigger than 0."},{"Start":"03:08.285 ","End":"03:12.500","Text":"Often won\u0027t bother with the domain of the transform function."},{"Start":"03:12.500 ","End":"03:13.849","Text":"But to be precise,"},{"Start":"03:13.849 ","End":"03:18.425","Text":"we should say that this transform applies when s is positive."},{"Start":"03:18.425 ","End":"03:25.040","Text":"In our second example will show that the Laplace transform of the function e^t,"},{"Start":"03:25.040 ","End":"03:29.425","Text":"the exponential function is 1 over s minus 1."},{"Start":"03:29.425 ","End":"03:31.035","Text":"Let me demonstrate this."},{"Start":"03:31.035 ","End":"03:34.915","Text":"G(s) is the Laplace transform of the function e^t."},{"Start":"03:34.915 ","End":"03:36.635","Text":"If you look at the definition,"},{"Start":"03:36.635 ","End":"03:41.840","Text":"everything\u0027s the same except this part which is where we put our particular function,"},{"Start":"03:41.840 ","End":"03:48.665","Text":"g and now we have to do integration with respect to t using the algebra of exponents,"},{"Start":"03:48.665 ","End":"03:53.180","Text":"we can rewrite this as 1t minus st which is"},{"Start":"03:53.180 ","End":"03:58.190","Text":"1 minus st. What we have is the integral of some constant times t. I mean,"},{"Start":"03:58.190 ","End":"04:00.035","Text":"as far as t goes, this is the constant,"},{"Start":"04:00.035 ","End":"04:04.250","Text":"which is just the exponent divided by this constant 1 minus s,"},{"Start":"04:04.250 ","End":"04:06.335","Text":"taken from 0 to infinity."},{"Start":"04:06.335 ","End":"04:07.730","Text":"We\u0027re plugging in infinity,"},{"Start":"04:07.730 ","End":"04:11.870","Text":"but really plugging in a very large number and taking the limit to infinity."},{"Start":"04:11.870 ","End":"04:17.220","Text":"This integral will converge for s bigger than 1,"},{"Start":"04:17.220 ","End":"04:21.530","Text":"in which case 1 minus s is negative and so a negative number times"},{"Start":"04:21.530 ","End":"04:26.300","Text":"t as t goes to infinity will be 0 and when t is 0,"},{"Start":"04:26.300 ","End":"04:28.855","Text":"we just get 1 over 1 minus s,"},{"Start":"04:28.855 ","End":"04:35.000","Text":"so we get 0 minus 1 over 1 minus s. This is what we get another say."},{"Start":"04:35.000 ","End":"04:36.980","Text":"This will be defined for s bigger than 1,"},{"Start":"04:36.980 ","End":"04:43.280","Text":"although usually we won\u0027t be pedantic about writing the domain for s and of course,"},{"Start":"04:43.280 ","End":"04:47.375","Text":"there\u0027s plenty of solved examples after the tutorial."},{"Start":"04:47.375 ","End":"04:50.630","Text":"Now of course we don\u0027t want to keep doing the integral each"},{"Start":"04:50.630 ","End":"04:53.750","Text":"time there are certain functions that we often want."},{"Start":"04:53.750 ","End":"04:56.960","Text":"The Laplace transform of like 1 or like"},{"Start":"04:56.960 ","End":"05:00.440","Text":"e^t in our examples and someone\u0027s done them already,"},{"Start":"05:00.440 ","End":"05:02.520","Text":"so there\u0027s no point doing them again and again,"},{"Start":"05:02.520 ","End":"05:04.610","Text":"so they\u0027ve been put into a table."},{"Start":"05:04.610 ","End":"05:05.930","Text":"There are large tables,"},{"Start":"05:05.930 ","End":"05:08.555","Text":"but he is a small table, a partial table."},{"Start":"05:08.555 ","End":"05:11.450","Text":"Just to give you the idea of what a table looks like."},{"Start":"05:11.450 ","End":"05:15.200","Text":"On one column we have the function of t and on"},{"Start":"05:15.200 ","End":"05:20.620","Text":"the other side we will put the Laplace transform G(s),"},{"Start":"05:20.620 ","End":"05:22.770","Text":"and we just look it up in the table,"},{"Start":"05:22.770 ","End":"05:26.860","Text":"so here\u0027s our example of g(t) is 1,"},{"Start":"05:26.860 ","End":"05:29.690","Text":"G(s) is 1 over s we did this example."},{"Start":"05:29.690 ","End":"05:34.140","Text":"We also did this example in the particular case where a was 1,"},{"Start":"05:34.140 ","End":"05:35.295","Text":"we had e^t,"},{"Start":"05:35.295 ","End":"05:38.075","Text":"I\u0027m going to get 1 over s minus 1."},{"Start":"05:38.075 ","End":"05:42.190","Text":"As I say, sometimes you want to write the domain of"},{"Start":"05:42.190 ","End":"05:47.170","Text":"these here we had s bigger than 1 in our case and in general s bigger than a."},{"Start":"05:47.170 ","End":"05:50.550","Text":"But mostly we won\u0027t be that precise sum,"},{"Start":"05:50.550 ","End":"05:55.825","Text":"and we will omit this range for s. When you have an exam or a test,"},{"Start":"05:55.825 ","End":"06:00.385","Text":"supposedly you will be given a table larger than this one anyway."},{"Start":"06:00.385 ","End":"06:03.610","Text":"Now we talked about tables of Laplace transforms,"},{"Start":"06:03.610 ","End":"06:07.810","Text":"but that\u0027s not enough because sometimes we have a variation of what\u0027s"},{"Start":"06:07.810 ","End":"06:12.250","Text":"on the table and we need to know how to modify the transform."},{"Start":"06:12.250 ","End":"06:15.005","Text":"Let\u0027s look at the first rule and you\u0027ll understand."},{"Start":"06:15.005 ","End":"06:17.470","Text":"The first rule is written like this."},{"Start":"06:17.470 ","End":"06:20.290","Text":"The name of the rule is linearity."},{"Start":"06:20.290 ","End":"06:23.675","Text":"The Laplace transform is linear,"},{"Start":"06:23.675 ","End":"06:28.415","Text":"which means that if we have a linear combination of 2 functions, g and h,"},{"Start":"06:28.415 ","End":"06:31.235","Text":"a times 1 of them plus b times the other,"},{"Start":"06:31.235 ","End":"06:33.350","Text":"the one we take the transform of that,"},{"Start":"06:33.350 ","End":"06:36.500","Text":"we can break it up into 2 bits with a plus,"},{"Start":"06:36.500 ","End":"06:38.690","Text":"and we can take the constants out."},{"Start":"06:38.690 ","End":"06:43.250","Text":"For example, suppose we want Laplace transform of the function"},{"Start":"06:43.250 ","End":"06:47.795","Text":"4t^2 plus 10 sine t. You won\u0027t find this exactly in the table,"},{"Start":"06:47.795 ","End":"06:50.810","Text":"but you will find t^2 in the table on,"},{"Start":"06:50.810 ","End":"06:53.015","Text":"you\u0027ll find sine t in the table."},{"Start":"06:53.015 ","End":"06:56.330","Text":"According to the rule, this is what we would have."},{"Start":"06:56.330 ","End":"07:00.085","Text":"Now as I say, we can look up these 2 Laplace transforms."},{"Start":"07:00.085 ","End":"07:02.450","Text":"The actual answer is not the point here."},{"Start":"07:02.450 ","End":"07:05.360","Text":"But I\u0027ll tell you that if you look up t^2,"},{"Start":"07:05.360 ","End":"07:10.820","Text":"you get 2 over s^3 and if you look up sine t,"},{"Start":"07:10.820 ","End":"07:15.410","Text":"you get 1 over s^2 plus 1."},{"Start":"07:15.410 ","End":"07:18.980","Text":"I\u0027m going to use the computation 4 times this plus 10 times this."},{"Start":"07:18.980 ","End":"07:23.300","Text":"That\u0027s not the point. The point is that we can split it up as follows."},{"Start":"07:23.300 ","End":"07:28.370","Text":"The second rule is that if we have some known function,"},{"Start":"07:28.370 ","End":"07:31.685","Text":"maybe this function g(t) is in the table,"},{"Start":"07:31.685 ","End":"07:32.960","Text":"but we don\u0027t have this."},{"Start":"07:32.960 ","End":"07:34.940","Text":"We have some power of t,"},{"Start":"07:34.940 ","End":"07:38.000","Text":"like t^4 times some unknown function."},{"Start":"07:38.000 ","End":"07:42.590","Text":"Then what we do is we look at g(t) in the table and we"},{"Start":"07:42.590 ","End":"07:47.990","Text":"take the Laplace transform together on L. Laplace transform of it,"},{"Start":"07:47.990 ","End":"07:55.550","Text":"and then this means the nth derivative of this function to differentiate n times."},{"Start":"07:55.550 ","End":"08:01.940","Text":"Then you multiply by a plus or a minus according to what n is and is even,"},{"Start":"08:01.940 ","End":"08:03.185","Text":"it\u0027ll come out plus,"},{"Start":"08:03.185 ","End":"08:05.270","Text":"and there\u0027s odd, it\u0027ll come out minus."},{"Start":"08:05.270 ","End":"08:07.085","Text":"If we apply this rule,"},{"Start":"08:07.085 ","End":"08:11.900","Text":"we get the Laplace transform of t^3 sine 4t,"},{"Start":"08:11.900 ","End":"08:13.715","Text":"which you won\u0027t find in the table,"},{"Start":"08:13.715 ","End":"08:16.388","Text":"is gong to be minus 1^3,"},{"Start":"08:16.388 ","End":"08:18.680","Text":"then the Laplace transform of sine 4t."},{"Start":"08:18.680 ","End":"08:21.725","Text":"But this has to be differentiated 3 times."},{"Start":"08:21.725 ","End":"08:23.420","Text":"Not going to do the whole computation,"},{"Start":"08:23.420 ","End":"08:33.229","Text":"but sine 4t gives us look at the table 4 over s^2 plus 4^2, which is 16."},{"Start":"08:33.229 ","End":"08:36.380","Text":"But then we have to take the third derivative."},{"Start":"08:36.380 ","End":"08:38.090","Text":"This is prime, prime prime,"},{"Start":"08:38.090 ","End":"08:39.830","Text":"that meaning third derivative of this,"},{"Start":"08:39.830 ","End":"08:42.295","Text":"and then multiply it by minus 1."},{"Start":"08:42.295 ","End":"08:43.760","Text":"Just in case you\u0027re wondering,"},{"Start":"08:43.760 ","End":"08:44.840","Text":"I wanted to try it yourself."},{"Start":"08:44.840 ","End":"08:48.485","Text":"The third derivative of this comes out to be this,"},{"Start":"08:48.485 ","End":"08:51.170","Text":"minus 1^3 is minus."},{"Start":"08:51.170 ","End":"08:58.275","Text":"The final answer would be minus with the minus and there we are the transform of this."},{"Start":"08:58.275 ","End":"09:01.715","Text":"Now I want to talk about the types of exercises,"},{"Start":"09:01.715 ","End":"09:03.125","Text":"and there\u0027s quite a few of them,"},{"Start":"09:03.125 ","End":"09:09.005","Text":"and they\u0027re sorted according to different types of exercises to compute"},{"Start":"09:09.005 ","End":"09:12.590","Text":"the transform of certain function using"},{"Start":"09:12.590 ","End":"09:16.505","Text":"the table and the properties of the transform that we gave,"},{"Start":"09:16.505 ","End":"09:18.035","Text":"we have 2 properties."},{"Start":"09:18.035 ","End":"09:23.030","Text":"After that, they\u0027ll be a couple of exercises where we have to compute"},{"Start":"09:23.030 ","End":"09:28.954","Text":"the transform from the definition and it\u0027s usually will be piecewise functions."},{"Start":"09:28.954 ","End":"09:30.545","Text":"I mean, for example,"},{"Start":"09:30.545 ","End":"09:37.020","Text":"that we have g(t) is equal to t^2 when t"},{"Start":"09:37.020 ","End":"09:43.700","Text":"is less than 1 and the constant function 1 when t is bigger or equal to 1,"},{"Start":"09:43.700 ","End":"09:46.580","Text":"for example, then we can\u0027t use the table for this,"},{"Start":"09:46.580 ","End":"09:49.430","Text":"we have to compute it directly."},{"Start":"09:49.430 ","End":"09:52.610","Text":"Then we\u0027ll have some exercises computing the transform of"},{"Start":"09:52.610 ","End":"09:57.995","Text":"a periodic function and I\u0027ll explain in the exercise while the periodic function is."},{"Start":"09:57.995 ","End":"09:59.960","Text":"But for example, sine t is"},{"Start":"09:59.960 ","End":"10:04.090","Text":"a periodic function because it keeps repeating itself every 2Pi."},{"Start":"10:04.090 ","End":"10:08.030","Text":"Then finally, we\u0027re going to compute the transform of"},{"Start":"10:08.030 ","End":"10:12.055","Text":"a step function and I\u0027ll explain what a step function is,"},{"Start":"10:12.055 ","End":"10:17.990","Text":"or a function is expressed using the step function variations of the step function."},{"Start":"10:17.990 ","End":"10:22.655","Text":"Lastly, I just want to give you an overview of what this chapter contains."},{"Start":"10:22.655 ","End":"10:25.789","Text":"The chapter begins with the introduction,"},{"Start":"10:25.789 ","End":"10:29.405","Text":"which is this clip which we just did or are doing,"},{"Start":"10:29.405 ","End":"10:37.025","Text":"then the Laplace transform mostly just exercises of how to compute Laplace transforms."},{"Start":"10:37.025 ","End":"10:41.150","Text":"Then we\u0027ll move on to the inverse Laplace transform."},{"Start":"10:41.150 ","End":"10:45.215","Text":"The inverse just means going back from a function of s to a function of"},{"Start":"10:45.215 ","End":"10:50.735","Text":"t. If I take the Laplace transform of some function g(t) and I get G(s)."},{"Start":"10:50.735 ","End":"10:52.940","Text":"Then the inverse Laplace transform,"},{"Start":"10:52.940 ","End":"10:57.740","Text":"it\u0027s written like this with a minus 1 as a superscript and I apply"},{"Start":"10:57.740 ","End":"11:03.035","Text":"the inverse transform to G(s) I\u0027ve got to get back to g(t)."},{"Start":"11:03.035 ","End":"11:06.620","Text":"For instance, we showed that the constant function 1 has"},{"Start":"11:06.620 ","End":"11:12.140","Text":"the Laplace transform of 1 over s. If I apply the inverse Laplace transform to 1 over s,"},{"Start":"11:12.140 ","End":"11:15.010","Text":"I should get back to the function 1."},{"Start":"11:15.010 ","End":"11:18.140","Text":"Then this is the most important because we\u0027re in"},{"Start":"11:18.140 ","End":"11:21.785","Text":"the course of an ordinary differential equations,"},{"Start":"11:21.785 ","End":"11:29.930","Text":"so we\u0027ll show how to use the Laplace transform to solve certain differential equations,"},{"Start":"11:29.930 ","End":"11:34.290","Text":"that\u0027s it for the introduction and overview."}],"ID":7894}],"Thumbnail":null,"ID":4245},{"Name":"The Laplace Transform","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"1m 4s","ChapterTopicVideoID":7843,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.950","Text":"In this exercise, we want to compute the Laplace transform of t^2+ 4t-2."},{"Start":"00:05.950 ","End":"00:08.745","Text":"I\u0027ll switch to regular L for legibility."},{"Start":"00:08.745 ","End":"00:14.070","Text":"Anyway, we\u0027re going to use the linearity property of the Laplace transform."},{"Start":"00:14.070 ","End":"00:15.570","Text":"Just to jog your memory,"},{"Start":"00:15.570 ","End":"00:17.505","Text":"this is the linearity rule,"},{"Start":"00:17.505 ","End":"00:19.680","Text":"and it just means that we can break this up into"},{"Start":"00:19.680 ","End":"00:22.230","Text":"separate Laplace transforms for each t\u0027s,"},{"Start":"00:22.230 ","End":"00:25.320","Text":"and constants can come outside."},{"Start":"00:25.320 ","End":"00:31.110","Text":"At this point, we use the table of transforms to look up these three things,"},{"Start":"00:31.110 ","End":"00:35.160","Text":"and they all happen to be t to the power of something."},{"Start":"00:35.160 ","End":"00:39.050","Text":"The same rule can be used for all three pieces."},{"Start":"00:39.050 ","End":"00:41.195","Text":"Also, L(1),"},{"Start":"00:41.195 ","End":"00:44.720","Text":"which is really a special case of this is 1/s,"},{"Start":"00:44.720 ","End":"00:47.885","Text":"which is the same as if you plug in n=1 here."},{"Start":"00:47.885 ","End":"00:51.860","Text":"We need t and t^2 where n is going to be 1, and then 2,"},{"Start":"00:51.860 ","End":"00:54.560","Text":"so t^2 gives us this,"},{"Start":"00:54.560 ","End":"00:57.290","Text":"and t gives us this."},{"Start":"00:57.290 ","End":"01:04.530","Text":"Now I just want to slightly rearrange and this is the answer. We\u0027re done."}],"ID":7895},{"Watched":false,"Name":"Exercise 2","Duration":"1m 5s","ChapterTopicVideoID":7844,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.574","Text":"In this exercise, we want to compute the Laplace transform of this function."},{"Start":"00:04.574 ","End":"00:07.650","Text":"Now switch to regular L more legible."},{"Start":"00:07.650 ","End":"00:13.185","Text":"First one, I use the linearity rule for the transform."},{"Start":"00:13.185 ","End":"00:14.850","Text":"We break it up as follows."},{"Start":"00:14.850 ","End":"00:16.740","Text":"The 1/2 comes out in front here,"},{"Start":"00:16.740 ","End":"00:18.030","Text":"the 2 over root Pi,"},{"Start":"00:18.030 ","End":"00:20.654","Text":"and we get 3 separate terms."},{"Start":"00:20.654 ","End":"00:22.245","Text":"Now, each of these,"},{"Start":"00:22.245 ","End":"00:24.525","Text":"we can look it up in the table."},{"Start":"00:24.525 ","End":"00:26.477","Text":"This one and this one,"},{"Start":"00:26.477 ","End":"00:30.345","Text":"this is t to the 0 and t to the 4."},{"Start":"00:30.345 ","End":"00:34.020","Text":"I can use the t to the n with n is 0 or 4,"},{"Start":"00:34.020 ","End":"00:36.900","Text":"but this doesn\u0027t work for fractional powers."},{"Start":"00:36.900 ","End":"00:39.045","Text":"For root t, we need a separate rule,"},{"Start":"00:39.045 ","End":"00:41.100","Text":"and this is it."},{"Start":"00:41.100 ","End":"00:42.615","Text":"For t to the fourth,"},{"Start":"00:42.615 ","End":"00:46.290","Text":"we get this 4 factorial overs to the fifth,"},{"Start":"00:46.290 ","End":"00:51.930","Text":"1 we already know is 1 over s and root t from here is this."},{"Start":"00:51.930 ","End":"00:57.215","Text":"Then we just simplify this because some of the stuff will cancel."},{"Start":"00:57.215 ","End":"01:00.270","Text":"Full factorial is 24 over 2 is 12,"},{"Start":"01:00.270 ","End":"01:02.535","Text":"Pi cancels, 2 cancels,"},{"Start":"01:02.535 ","End":"01:05.880","Text":"and this is what we\u0027re left with. That\u0027s the answer."}],"ID":7896},{"Watched":false,"Name":"Exercise 3","Duration":"47s","ChapterTopicVideoID":7845,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have to compute the Laplace transform of this function."},{"Start":"00:04.650 ","End":"00:06.480","Text":"This is the curly L,"},{"Start":"00:06.480 ","End":"00:10.395","Text":"but I prefer to use a plane L, more legible."},{"Start":"00:10.395 ","End":"00:13.620","Text":"Now, notice that both of these are exponential,"},{"Start":"00:13.620 ","End":"00:17.190","Text":"but we have a sum and a constant times."},{"Start":"00:17.190 ","End":"00:20.895","Text":"We use the property of linearity to break it up,"},{"Start":"00:20.895 ","End":"00:24.735","Text":"the sum, and also to take the constant out of the second term."},{"Start":"00:24.735 ","End":"00:28.105","Text":"Now, each of these is e to the something t,"},{"Start":"00:28.105 ","End":"00:30.350","Text":"and then the table we have this,"},{"Start":"00:30.350 ","End":"00:35.750","Text":"which is good for both bits because we can take 1 time a to be minus 4,"},{"Start":"00:35.750 ","End":"00:39.335","Text":"and then the other case we can take a to be 2."},{"Start":"00:39.335 ","End":"00:41.540","Text":"Notice that it\u0027s a minus a here,"},{"Start":"00:41.540 ","End":"00:45.844","Text":"so s minus a is s plus 4 here and s minus 2 here."},{"Start":"00:45.844 ","End":"00:48.120","Text":"This is the answer."}],"ID":7897},{"Watched":false,"Name":"Exercise 4","Duration":"1m 15s","ChapterTopicVideoID":7846,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Here we have to compute the Laplace transform."},{"Start":"00:02.820 ","End":"00:07.455","Text":"That is the curly L of the hyperbolic cosine of 4t."},{"Start":"00:07.455 ","End":"00:09.840","Text":"I\u0027m going to switch to irregular L. Now,"},{"Start":"00:09.840 ","End":"00:13.305","Text":"I want to remind you what the cosine hyperbolic is."},{"Start":"00:13.305 ","End":"00:15.555","Text":"Some people pronounce this coshine."},{"Start":"00:15.555 ","End":"00:18.915","Text":"Anyway, it\u0027s equal to the following."},{"Start":"00:18.915 ","End":"00:23.865","Text":"In our case, we wanted to substitute instead of x for t,"},{"Start":"00:23.865 ","End":"00:26.990","Text":"and so we want the Laplace transform of this."},{"Start":"00:26.990 ","End":"00:31.380","Text":"Now, we\u0027re going to use the linearity of the Laplace transform."},{"Start":"00:31.380 ","End":"00:32.730","Text":"In case you\u0027ve forgotten,"},{"Start":"00:32.730 ","End":"00:35.460","Text":"this is the rule for linearity."},{"Start":"00:35.460 ","End":"00:39.380","Text":"In our case, it means that we can take the half outside the brackets and then we can"},{"Start":"00:39.380 ","End":"00:43.775","Text":"also apply the Laplace transform separately to each of the bit,"},{"Start":"00:43.775 ","End":"00:45.595","Text":"so we end up with this."},{"Start":"00:45.595 ","End":"00:50.630","Text":"Then we use this formula from the table, table of transforms."},{"Start":"00:50.630 ","End":"00:58.480","Text":"But one time we\u0027ll want a to equal 4 and then the other case we want A to be minus 4."},{"Start":"00:58.480 ","End":"01:00.665","Text":"This is what we get."},{"Start":"01:00.665 ","End":"01:04.490","Text":"We could say this is the final answer or you could simplify it."},{"Start":"01:04.490 ","End":"01:05.990","Text":"If you simplify it,"},{"Start":"01:05.990 ","End":"01:12.200","Text":"I think that what you get is s over s^2 minus 16,"},{"Start":"01:12.200 ","End":"01:15.720","Text":"but this is good enough. We\u0027re done."}],"ID":7898},{"Watched":false,"Name":"Exercise 5","Duration":"1m 3s","ChapterTopicVideoID":7847,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we have to compute the Laplace transform,"},{"Start":"00:03.750 ","End":"00:08.535","Text":"that\u0027s this curly L of the hyperbolic sine(10t)."},{"Start":"00:08.535 ","End":"00:10.035","Text":"I prefer a regular L,"},{"Start":"00:10.035 ","End":"00:12.930","Text":"and I\u0027ll remind you what the hyperbolic sine is."},{"Start":"00:12.930 ","End":"00:15.735","Text":"It\u0027s sometimes pronounced sinh."},{"Start":"00:15.735 ","End":"00:19.650","Text":"Sinh(x) is a 1/2(e^x minus e^x),"},{"Start":"00:19.650 ","End":"00:20.895","Text":"and in our case,"},{"Start":"00:20.895 ","End":"00:24.210","Text":"we need to substitute x=10t."},{"Start":"00:24.210 ","End":"00:25.950","Text":"So instead of this,"},{"Start":"00:25.950 ","End":"00:28.695","Text":"we have to look up the Laplace transform of this,"},{"Start":"00:28.695 ","End":"00:31.125","Text":"and now we use the linearity property,"},{"Start":"00:31.125 ","End":"00:37.275","Text":"so we can take the half outside and apply the transform to each term separately here."},{"Start":"00:37.275 ","End":"00:39.305","Text":"From the lookup table,"},{"Start":"00:39.305 ","End":"00:42.800","Text":"we get this formula and we\u0027re going to use it twice."},{"Start":"00:42.800 ","End":"00:45.110","Text":"Once with a=10,"},{"Start":"00:45.110 ","End":"00:49.945","Text":"for this bit, and once with a=-10, for this bit,"},{"Start":"00:49.945 ","End":"00:54.169","Text":"and so this is our answer though if you want to do algebraic manipulation,"},{"Start":"00:54.169 ","End":"01:00.590","Text":"it does simplify to 10/S^2 minus a 100."},{"Start":"01:00.590 ","End":"01:03.570","Text":"But you could leave the answer like this."}],"ID":7899},{"Watched":false,"Name":"Exercise 6","Duration":"1m 18s","ChapterTopicVideoID":7848,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.910","Text":"This curly L means that we have to compute the Laplace transform of this function,"},{"Start":"00:05.910 ","End":"00:10.890","Text":"but we don\u0027t find anything like this in the table."},{"Start":"00:10.890 ","End":"00:14.309","Text":"If you don\u0027t mind, I\u0027m going to use regular L. Anyway,"},{"Start":"00:14.309 ","End":"00:15.540","Text":"what to do about this,"},{"Start":"00:15.540 ","End":"00:20.466","Text":"there\u0027s no rule for the Laplace transform of a product like we have for a sum L,"},{"Start":"00:20.466 ","End":"00:23.895","Text":"the answer is to use trigonometric identities."},{"Start":"00:23.895 ","End":"00:27.161","Text":"The one I have in mind is this identity,"},{"Start":"00:27.161 ","End":"00:29.700","Text":"because I see there\u0027s a sine times a cosine."},{"Start":"00:29.700 ","End":"00:35.575","Text":"There\u0027s a 2 here, but we can adjust that so if we let Alpha=2t,"},{"Start":"00:35.575 ","End":"00:39.135","Text":"we get this and now we\u0027re getting very close."},{"Start":"00:39.135 ","End":"00:42.930","Text":"If I put the 2 on the other side and make it 1/2,"},{"Start":"00:42.930 ","End":"00:48.260","Text":"then our original function is the same as 1/2 sine 4t."},{"Start":"00:48.260 ","End":"00:50.255","Text":"This is easier to solve,"},{"Start":"00:50.255 ","End":"01:00.390","Text":"we have to take the 1/2 out first using linearity like so."},{"Start":"01:00.390 ","End":"01:04.079","Text":"Now, I produced from the table this formula,"},{"Start":"01:04.079 ","End":"01:06.750","Text":"and in our case a=4,"},{"Start":"01:06.750 ","End":"01:09.015","Text":"and so this is what we get this as the 1/2."},{"Start":"01:09.015 ","End":"01:13.530","Text":"This is the a over s^2 plus a^2, 4^2 is 16."},{"Start":"01:13.530 ","End":"01:15.510","Text":"The 1/2 and the 4 give us a 2,"},{"Start":"01:15.510 ","End":"01:18.250","Text":"so this is the answer."}],"ID":7900},{"Watched":false,"Name":"Exercise 7","Duration":"1m 35s","ChapterTopicVideoID":7849,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"This curly L means that we have to find"},{"Start":"00:02.490 ","End":"00:07.725","Text":"the Laplace transform of this function, sin2t cos3t."},{"Start":"00:07.725 ","End":"00:10.650","Text":"I\u0027m going use a regular L if that\u0027s okay with you."},{"Start":"00:10.650 ","End":"00:13.530","Text":"This function isn\u0027t to be found in the table"},{"Start":"00:13.530 ","End":"00:16.590","Text":"and there\u0027s no rule for a product with Laplace transforms."},{"Start":"00:16.590 ","End":"00:20.430","Text":"What we\u0027ll have to do is trigonometric identities."},{"Start":"00:20.430 ","End":"00:23.025","Text":"This is the one I have in mind,"},{"Start":"00:23.025 ","End":"00:29.415","Text":"with alpha equals 2t and beta equals 3t."},{"Start":"00:29.415 ","End":"00:34.380","Text":"Then alpha plus beta is 5t and alpha minus beta is"},{"Start":"00:34.380 ","End":"00:39.500","Text":"minus t. But the left-hand side here is not exactly the function we want."},{"Start":"00:39.500 ","End":"00:41.750","Text":"There\u0027s an extra 2 here, so no problem."},{"Start":"00:41.750 ","End":"00:43.760","Text":"I\u0027ll just bring it to the other side,"},{"Start":"00:43.760 ","End":"00:46.490","Text":"put brackets and now our problem"},{"Start":"00:46.490 ","End":"00:49.250","Text":"becomes defined the Laplace transform of this expression,"},{"Start":"00:49.250 ","End":"00:50.495","Text":"which will be easier."},{"Start":"00:50.495 ","End":"00:53.450","Text":"We\u0027ll be able to take the 1/2 out using"},{"Start":"00:53.450 ","End":"00:58.115","Text":"the linearity and to break it up and this is what we get."},{"Start":"00:58.115 ","End":"01:03.190","Text":"Now, we do have Laplace transform of sine in the table."},{"Start":"01:03.190 ","End":"01:06.965","Text":"I forgot to mention something earlier that this minus t,"},{"Start":"01:06.965 ","End":"01:09.260","Text":"I took the minus out because sine is"},{"Start":"01:09.260 ","End":"01:12.610","Text":"an odd function so I can put the minus in front of the sine."},{"Start":"01:12.610 ","End":"01:13.955","Text":"Yeah, I forgot to say that."},{"Start":"01:13.955 ","End":"01:15.840","Text":"Back to the table,"},{"Start":"01:15.840 ","End":"01:19.520","Text":"the table has the sine of at in general."},{"Start":"01:19.520 ","End":"01:25.405","Text":"We will take it once with a=5 and the other time with a=1,"},{"Start":"01:25.405 ","End":"01:28.250","Text":"and we end up with this expression,"},{"Start":"01:28.250 ","End":"01:30.800","Text":"which could probably be simplified,"},{"Start":"01:30.800 ","End":"01:34.950","Text":"but we\u0027ll leave it at that. Okay, done."}],"ID":7901},{"Watched":false,"Name":"Exercise 8","Duration":"1m 14s","ChapterTopicVideoID":7850,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.209","Text":"In this exercise, we have to find the Laplace transform."},{"Start":"00:03.209 ","End":"00:07.105","Text":"That\u0027s the curly L of sin^2 t. I prefer"},{"Start":"00:07.105 ","End":"00:11.308","Text":"regular L. Now there\u0027s no entry in the table for sin^2 t,"},{"Start":"00:11.308 ","End":"00:15.405","Text":"so we\u0027ll have to use some trigonometric identities."},{"Start":"00:15.405 ","End":"00:21.645","Text":"What I had in mind was this one can put the 1/2 Inside the brackets."},{"Start":"00:21.645 ","End":"00:27.045","Text":"This is essentially the same as this and this is easier to compute."},{"Start":"00:27.045 ","End":"00:29.250","Text":"From here I\u0027m going to use linearity."},{"Start":"00:29.250 ","End":"00:31.830","Text":"This is the summary of what it means. We won\u0027t go into it."},{"Start":"00:31.830 ","End":"00:35.930","Text":"What it means in our case is that we can take the 1/2 outside"},{"Start":"00:35.930 ","End":"00:39.830","Text":"the brackets and the wind also break up the minus like this."},{"Start":"00:39.830 ","End":"00:43.430","Text":"We end up with 1/2 the Laplace transform of the Function 1,"},{"Start":"00:43.430 ","End":"00:48.875","Text":"not the number 1, the constant function 1 minus 1/2 Laplace transform of cos 2t."},{"Start":"00:48.875 ","End":"00:56.210","Text":"Now both these, the one on the cos 2t appear in the table of Laplace transforms."},{"Start":"00:56.210 ","End":"00:58.670","Text":"This one exactly as it says on this one, well,"},{"Start":"00:58.670 ","End":"01:03.940","Text":"we\u0027ll just have to make a=2 in the second formula."},{"Start":"01:03.940 ","End":"01:05.690","Text":"If we plug those in here,"},{"Start":"01:05.690 ","End":"01:11.630","Text":"what we get is 1/2 times 1/s and 1/2 times s over s^2 plus 2^2,"},{"Start":"01:11.630 ","End":"01:15.270","Text":"which is 4. That\u0027s the answer."}],"ID":7902},{"Watched":false,"Name":"Exercise 9","Duration":"1m 11s","ChapterTopicVideoID":7851,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.749","Text":"In this exercise, we have to compute the Laplace transform,"},{"Start":"00:03.749 ","End":"00:07.170","Text":"that\u0027s this curly L of cos^2 4t."},{"Start":"00:07.170 ","End":"00:10.545","Text":"I prefer to use irregular L. Anyway,"},{"Start":"00:10.545 ","End":"00:13.440","Text":"we look for something in the table of Laplace"},{"Start":"00:13.440 ","End":"00:16.935","Text":"transform that\u0027s similar to this and we don\u0027t find one,"},{"Start":"00:16.935 ","End":"00:19.560","Text":"so we need trigonometric identities."},{"Start":"00:19.560 ","End":"00:21.210","Text":"This is the one I had in mind."},{"Start":"00:21.210 ","End":"00:23.625","Text":"Sometimes it\u0027s written slightly differently,"},{"Start":"00:23.625 ","End":"00:26.220","Text":"more commonly seen in this form."},{"Start":"00:26.220 ","End":"00:28.965","Text":"Anyway, it\u0027s the same thing."},{"Start":"00:28.965 ","End":"00:34.290","Text":"If we let Alpha equals 4t in this formula,"},{"Start":"00:34.290 ","End":"00:35.655","Text":"we get this,"},{"Start":"00:35.655 ","End":"00:37.920","Text":"and this is easier to compute."},{"Start":"00:37.920 ","End":"00:41.730","Text":"First of all, we\u0027ll use the linearity of the transform."},{"Start":"00:41.730 ","End":"00:43.784","Text":"That gives us this,"},{"Start":"00:43.784 ","End":"00:45.855","Text":"where we have two functions."},{"Start":"00:45.855 ","End":"00:47.565","Text":"One is the Function 1,"},{"Start":"00:47.565 ","End":"00:50.155","Text":"constant function and cos 8t."},{"Start":"00:50.155 ","End":"00:55.745","Text":"Now, this is in the table and so is this in a bit more general form."},{"Start":"00:55.745 ","End":"01:00.815","Text":"This is the transform of 1 and this is the transform of cosine,"},{"Start":"01:00.815 ","End":"01:05.120","Text":"well, not 8t, But if we let a=8 here, we\u0027ll have that."},{"Start":"01:05.120 ","End":"01:08.645","Text":"If we plug it all in and compute 8^2 is 64,"},{"Start":"01:08.645 ","End":"01:12.029","Text":"then this is the answer."}],"ID":7903},{"Watched":false,"Name":"Exercise 10","Duration":"1m 31s","ChapterTopicVideoID":7852,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this exercise we need to compute the Laplace transform"},{"Start":"00:04.140 ","End":"00:08.445","Text":"that\u0027s indicated by this curly L(t^2 sin 4t)."},{"Start":"00:08.445 ","End":"00:11.950","Text":"But I\u0027ll use the regular L, easier."},{"Start":"00:11.960 ","End":"00:14.070","Text":"At least in my table,"},{"Start":"00:14.070 ","End":"00:19.455","Text":"I don\u0027t have the Laplace transform (t^2 sin 4t) or something similar to it."},{"Start":"00:19.455 ","End":"00:23.369","Text":"But we do have one of the rules of Laplace transforms,"},{"Start":"00:23.369 ","End":"00:28.935","Text":"and this is the following rule where the Laplace transform of"},{"Start":"00:28.935 ","End":"00:35.235","Text":"a power of t times some function is minus 1 to the n. Now,"},{"Start":"00:35.235 ","End":"00:39.330","Text":"G(s) is the Laplace transform of g(t)."},{"Start":"00:39.330 ","End":"00:42.870","Text":"This n in brackets means nth derivative."},{"Start":"00:42.870 ","End":"00:51.585","Text":"In our case we\u0027re letting n=2 and g(t) will be sin 4t,"},{"Start":"00:51.585 ","End":"00:57.525","Text":"and we can get G(s) from this formula with"},{"Start":"00:57.525 ","End":"01:06.975","Text":"a=4 so my G(s) is this with a=4 which is 4 over s^2 plus 16, 16 is 4^2."},{"Start":"01:06.975 ","End":"01:14.150","Text":"This bit is minus 1^2 and this 2 in brackets means second derivative,"},{"Start":"01:14.150 ","End":"01:16.220","Text":"so I write it as prime, prime."},{"Start":"01:16.220 ","End":"01:17.870","Text":"Now, minus 1^2 is 1,"},{"Start":"01:17.870 ","End":"01:19.475","Text":"so I don\u0027t need that."},{"Start":"01:19.475 ","End":"01:21.485","Text":"The second derivative of this,"},{"Start":"01:21.485 ","End":"01:24.620","Text":"if you compute it comes out to be this."},{"Start":"01:24.620 ","End":"01:26.690","Text":"I\u0027ll spare you the details."},{"Start":"01:26.690 ","End":"01:29.330","Text":"That\u0027s the answer, may be it can be simplified."},{"Start":"01:29.330 ","End":"01:32.040","Text":"I don\u0027t know, but we\u0027ll leave it at that."}],"ID":7904},{"Watched":false,"Name":"Exercise 11","Duration":"1m 58s","ChapterTopicVideoID":7853,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have to compute the Laplace transform,"},{"Start":"00:03.570 ","End":"00:08.025","Text":"that\u0027s indicated by this curly L of t^4th e^2t."},{"Start":"00:08.025 ","End":"00:10.905","Text":"I think I\u0027ll just switch to plain L. Now,"},{"Start":"00:10.905 ","End":"00:17.370","Text":"in some tables you might find an entry for t^n e^at,"},{"Start":"00:17.370 ","End":"00:19.650","Text":"but not in this simpler tables."},{"Start":"00:19.650 ","End":"00:22.425","Text":"In any event, I\u0027m not going to use the formula for this."},{"Start":"00:22.425 ","End":"00:26.970","Text":"I\u0027m going to use the rule for t^n times a function."},{"Start":"00:26.970 ","End":"00:29.325","Text":"This is the rule I\u0027m referring to,"},{"Start":"00:29.325 ","End":"00:30.780","Text":"where we have t^n."},{"Start":"00:30.780 ","End":"00:37.055","Text":"Well, n will equal 4 and g of t will be the function e^2t."},{"Start":"00:37.055 ","End":"00:41.383","Text":"G(s) is the transform of g(t),"},{"Start":"00:41.383 ","End":"00:46.910","Text":"and that we do have in every table they will give you the transform of this."},{"Start":"00:46.910 ","End":"00:50.580","Text":"Well, not specifically to the 2t but e^at,"},{"Start":"00:50.580 ","End":"00:54.425","Text":"we can let a=2 in this formula."},{"Start":"00:54.425 ","End":"00:58.010","Text":"Remember this n in brackets means derivative,"},{"Start":"00:58.010 ","End":"00:59.420","Text":"like if n is 4,"},{"Start":"00:59.420 ","End":"01:01.745","Text":"this means the 4th derivative."},{"Start":"01:01.745 ","End":"01:05.620","Text":"What we end up with is minus 1^4."},{"Start":"01:05.620 ","End":"01:07.890","Text":"Then we have this 1 overs minus a,"},{"Start":"01:07.890 ","End":"01:09.870","Text":"which is 1 over s minus 2."},{"Start":"01:09.870 ","End":"01:13.535","Text":"4th derivative, I just wrote it as \u0027\u0027\u0027."},{"Start":"01:13.535 ","End":"01:15.470","Text":"Now, minus 1 to the 4th is 1,"},{"Start":"01:15.470 ","End":"01:16.805","Text":"so we don\u0027t need that,"},{"Start":"01:16.805 ","End":"01:20.065","Text":"and if you differentiate this 4 times,"},{"Start":"01:20.065 ","End":"01:22.595","Text":"then we get this."},{"Start":"01:22.595 ","End":"01:27.665","Text":"That\u0027s the answer. But if you wanted to do it with this formula,"},{"Start":"01:27.665 ","End":"01:29.034","Text":"I\u0027ll show you,"},{"Start":"01:29.034 ","End":"01:37.590","Text":"this gives us n factorial over s minus a^n plus 1."},{"Start":"01:37.590 ","End":"01:45.165","Text":"If we let n equals 4 and a equals 2,"},{"Start":"01:45.165 ","End":"01:48.765","Text":"then we get 4 factorial is 24,"},{"Start":"01:48.765 ","End":"01:54.780","Text":"s minus 2 and 4 plus 1 is 5."},{"Start":"01:54.780 ","End":"01:56.355","Text":"Same answer."},{"Start":"01:56.355 ","End":"01:58.720","Text":"Anyway, we\u0027re done."}],"ID":7905},{"Watched":false,"Name":"Exercise 12","Duration":"37s","ChapterTopicVideoID":7854,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to compute the Laplace transform,"},{"Start":"00:03.450 ","End":"00:07.245","Text":"that\u0027s this symbol of e^2t sin 4t."},{"Start":"00:07.245 ","End":"00:09.195","Text":"I\u0027ll use a regular L. Now,"},{"Start":"00:09.195 ","End":"00:14.475","Text":"there is an entry in the table of Laplace transforms that looks like this."},{"Start":"00:14.475 ","End":"00:17.160","Text":"This is what that formula looks like."},{"Start":"00:17.160 ","End":"00:21.405","Text":"What we have to do is put the appropriate a and b here."},{"Start":"00:21.405 ","End":"00:23.460","Text":"In our case, b is minus 2."},{"Start":"00:23.460 ","End":"00:26.865","Text":"Note the minus because the formula is e^-bt,"},{"Start":"00:26.865 ","End":"00:29.595","Text":"so to get plus 2t I need minus 2."},{"Start":"00:29.595 ","End":"00:31.110","Text":"There is 4,"},{"Start":"00:31.110 ","End":"00:32.715","Text":"just plug it in."},{"Start":"00:32.715 ","End":"00:37.510","Text":"Remember that 4^2 is 16 and this is the answer."}],"ID":7906},{"Watched":false,"Name":"Exercise 13","Duration":"3m 48s","ChapterTopicVideoID":7855,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this exercise, we have to compute the Laplace transform of this function."},{"Start":"00:05.460 ","End":"00:07.680","Text":"Which is defined piecewise,"},{"Start":"00:07.680 ","End":"00:09.825","Text":"in two pieces to be precise,"},{"Start":"00:09.825 ","End":"00:14.955","Text":"and it\u0027s defined one way to when t is less than or equal to 1,"},{"Start":"00:14.955 ","End":"00:17.400","Text":"and it\u0027s defined another way when t is bigger than 1,"},{"Start":"00:17.400 ","End":"00:19.770","Text":"here it\u0027s t here it\u0027s 1."},{"Start":"00:19.770 ","End":"00:23.445","Text":"We can\u0027t just look it up in the table."},{"Start":"00:23.445 ","End":"00:28.620","Text":"In these cases, we pretty much have to do it from the definition i.e.,"},{"Start":"00:28.620 ","End":"00:30.555","Text":"we have to compute an integral."},{"Start":"00:30.555 ","End":"00:34.270","Text":"This is the definition of the Laplace transform of g,"},{"Start":"00:34.270 ","End":"00:39.605","Text":"but we can\u0027t just substitute g as usual because it\u0027s piecewise."},{"Start":"00:39.605 ","End":"00:44.690","Text":"What we have to do is break this range from 0 to infinity into 2 parts;"},{"Start":"00:44.690 ","End":"00:47.530","Text":"from 0-1 and from 1 to infinity."},{"Start":"00:47.530 ","End":"00:54.300","Text":"From 0-1, g of t is defined to be t,"},{"Start":"00:54.300 ","End":"00:56.000","Text":"and from 1 to infinity,"},{"Start":"00:56.000 ","End":"00:58.018","Text":"it\u0027s defined to be 1."},{"Start":"00:58.018 ","End":"01:00.890","Text":"That\u0027s why we have to split it up."},{"Start":"01:00.890 ","End":"01:03.680","Text":"Now I don\u0027t want to break the flow by computing integrals,"},{"Start":"01:03.680 ","End":"01:05.490","Text":"but we\u0027ll do those at the end."},{"Start":"01:05.490 ","End":"01:08.030","Text":"I\u0027ll just tell you that the answer to this one is this,"},{"Start":"01:08.030 ","End":"01:09.790","Text":"and the answer to this one is this,"},{"Start":"01:09.790 ","End":"01:14.075","Text":"and I owe you at the end to do these integrals."},{"Start":"01:14.075 ","End":"01:16.460","Text":"A bit of simplification,"},{"Start":"01:16.460 ","End":"01:18.410","Text":"a common denominator, s^2."},{"Start":"01:18.410 ","End":"01:23.105","Text":"This becomes se to the minus s. Then when we collect stuff together,"},{"Start":"01:23.105 ","End":"01:24.710","Text":"this is the answer we get."},{"Start":"01:24.710 ","End":"01:28.475","Text":"Now I have to show you how I did these 2 integrals."},{"Start":"01:28.475 ","End":"01:31.820","Text":"Now, the indefinite integral of e to the minus t times"},{"Start":"01:31.820 ","End":"01:35.690","Text":"t. We can do the integration by parts,"},{"Start":"01:35.690 ","End":"01:39.530","Text":"or you might even find it in the table of integrals."},{"Start":"01:39.530 ","End":"01:41.735","Text":"I\u0027m not going to go into all the details,"},{"Start":"01:41.735 ","End":"01:44.000","Text":"but then there\u0027s the indefinite integral."},{"Start":"01:44.000 ","End":"01:48.600","Text":"Now we have to substitute the limits 0 and 1."},{"Start":"01:49.670 ","End":"01:52.080","Text":"Well this is t equals 1,"},{"Start":"01:52.080 ","End":"01:53.160","Text":"of course not s,"},{"Start":"01:53.160 ","End":"01:54.675","Text":"I\u0027ll just emphasize that."},{"Start":"01:54.675 ","End":"01:56.050","Text":"When t is 1,"},{"Start":"01:56.050 ","End":"01:58.100","Text":"you can see that we get this."},{"Start":"01:58.100 ","End":"02:03.395","Text":"When t is 0, this part just disappears because it\u0027s 1,"},{"Start":"02:03.395 ","End":"02:07.640","Text":"and also this thing just becomes t is 0,"},{"Start":"02:07.640 ","End":"02:09.350","Text":"so this is just 1 also."},{"Start":"02:09.350 ","End":"02:12.095","Text":"We\u0027re just left with the minus 1 over s^2."},{"Start":"02:12.095 ","End":"02:15.965","Text":"Then we need to subtract the upper limit minus the lower limit."},{"Start":"02:15.965 ","End":"02:19.910","Text":"Then we simplify this minus times the other minus is a plus,"},{"Start":"02:19.910 ","End":"02:21.920","Text":"that\u0027s this 1 over s^2,"},{"Start":"02:21.920 ","End":"02:24.420","Text":"and the rest of it is the minus."},{"Start":"02:24.420 ","End":"02:27.660","Text":"It\u0027s this bit here over s^2,"},{"Start":"02:27.660 ","End":"02:29.749","Text":"and that\u0027s the first integral."},{"Start":"02:29.749 ","End":"02:32.315","Text":"The second integral I owe you is this one,"},{"Start":"02:32.315 ","End":"02:36.980","Text":"but this time it\u0027s an improper integral so it\u0027s a bit trickier."},{"Start":"02:36.980 ","End":"02:41.360","Text":"The indefinite integral is easier than before, it\u0027s straightforward."},{"Start":"02:41.360 ","End":"02:43.115","Text":"This is a function of t,"},{"Start":"02:43.115 ","End":"02:47.405","Text":"we just take this itself and divide by minus s,"},{"Start":"02:47.405 ","End":"02:50.300","Text":"which gives us minus 1 over s times this thing."},{"Start":"02:50.300 ","End":"02:54.140","Text":"Then I have to substitute 1 and infinity, because infinity,"},{"Start":"02:54.140 ","End":"02:56.120","Text":"I don\u0027t actually substitute infinity,"},{"Start":"02:56.120 ","End":"02:59.390","Text":"it\u0027s a limit of something large that goes to infinity."},{"Start":"02:59.390 ","End":"03:04.430","Text":"We\u0027re going to have to assume s is positive on the domain of this function,"},{"Start":"03:04.430 ","End":"03:06.275","Text":"then we have an integral."},{"Start":"03:06.275 ","End":"03:10.670","Text":"Now, if t goes to infinity and s is positive,"},{"Start":"03:10.670 ","End":"03:14.210","Text":"then minus st goes to minus infinity,"},{"Start":"03:14.210 ","End":"03:18.950","Text":"and the limit of e to the something and something goes to minus infinity is 0,"},{"Start":"03:18.950 ","End":"03:21.275","Text":"minus 0 is still 0."},{"Start":"03:21.275 ","End":"03:22.730","Text":"That\u0027s the upper limit."},{"Start":"03:22.730 ","End":"03:25.079","Text":"Now when we plug in 1, that\u0027s straightforward."},{"Start":"03:25.079 ","End":"03:27.890","Text":"Remember it\u0027s t that we\u0027re substituting so it just"},{"Start":"03:27.890 ","End":"03:31.235","Text":"looks like this because the t disappear because it\u0027s 1,"},{"Start":"03:31.235 ","End":"03:32.990","Text":"and then this minus,"},{"Start":"03:32.990 ","End":"03:35.405","Text":"minus is a plus."},{"Start":"03:35.405 ","End":"03:37.590","Text":"This is what we get."},{"Start":"03:37.590 ","End":"03:41.060","Text":"If you check these two results here,"},{"Start":"03:41.060 ","End":"03:44.285","Text":"we got in the previous page or what are declared,"},{"Start":"03:44.285 ","End":"03:47.940","Text":"now I\u0027ve shown you how I got to them."}],"ID":7907},{"Watched":false,"Name":"Exercise 14","Duration":"6m ","ChapterTopicVideoID":7856,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we need to compute the Laplace transform of this function,"},{"Start":"00:05.145 ","End":"00:07.680","Text":"and this is defined piecewise,"},{"Start":"00:07.680 ","End":"00:09.420","Text":"from 0 to 1,"},{"Start":"00:09.420 ","End":"00:10.680","Text":"it\u0027s defined as t,"},{"Start":"00:10.680 ","End":"00:13.530","Text":"and from 1 to infinity,"},{"Start":"00:13.530 ","End":"00:20.040","Text":"it\u0027s defined as 2 minus t. We can\u0027t use table of Laplace transforms,"},{"Start":"00:20.040 ","End":"00:23.595","Text":"we have to do this from the definition of the transform,"},{"Start":"00:23.595 ","End":"00:26.340","Text":"the integral, and this is how it\u0027s defined."},{"Start":"00:26.340 ","End":"00:29.910","Text":"We can\u0027t just plug g in because it\u0027s defined in two different ways."},{"Start":"00:29.910 ","End":"00:36.930","Text":"What we do is we break up the range from 0 to infinity into two ranges, from 0-1,"},{"Start":"00:36.930 ","End":"00:41.100","Text":"g(t) is defined as t, and that\u0027s this,"},{"Start":"00:41.100 ","End":"00:43.185","Text":"and from 1 to infinity,"},{"Start":"00:43.185 ","End":"00:46.740","Text":"it\u0027s defined as 2 minus t, which is this."},{"Start":"00:46.740 ","End":"00:51.380","Text":"I don\u0027t want to break the flow so I\u0027m going to tell you the answer to these integrals."},{"Start":"00:51.380 ","End":"00:52.940","Text":"This one gives me this,"},{"Start":"00:52.940 ","End":"00:54.409","Text":"this one gives me this,"},{"Start":"00:54.409 ","End":"00:56.480","Text":"and I owe you this at the end,"},{"Start":"00:56.480 ","End":"00:59.915","Text":"I will do these definite integrals at the end."},{"Start":"00:59.915 ","End":"01:03.385","Text":"Now, it\u0027s just a matter of tidying up."},{"Start":"01:03.385 ","End":"01:05.583","Text":"Both these fractions are over s^2,"},{"Start":"01:05.583 ","End":"01:12.143","Text":"so we\u0027ll just have to open this up and add this and then collect together,"},{"Start":"01:12.143 ","End":"01:13.880","Text":"and this is the answer."},{"Start":"01:13.880 ","End":"01:18.500","Text":"But we\u0027re not done because I still owe you these two integrals."},{"Start":"01:18.500 ","End":"01:20.890","Text":"The first one was this,"},{"Start":"01:20.890 ","End":"01:22.830","Text":"and I believe we\u0027ve seen this before,"},{"Start":"01:22.830 ","End":"01:25.610","Text":"the e to the minus st times t. If not,"},{"Start":"01:25.610 ","End":"01:27.320","Text":"this can be done by parts,"},{"Start":"01:27.320 ","End":"01:29.059","Text":"I\u0027m not going to do every detail."},{"Start":"01:29.059 ","End":"01:32.009","Text":"The in depth integral of this comes out to be this,"},{"Start":"01:32.009 ","End":"01:33.170","Text":"and to make it definite,"},{"Start":"01:33.170 ","End":"01:35.390","Text":"we substitute 0 and 1, well,"},{"Start":"01:35.390 ","End":"01:39.650","Text":"the 1 first and then subtract it from the 0 part anyway, you know what I mean."},{"Start":"01:39.650 ","End":"01:42.170","Text":"Plug in 1,"},{"Start":"01:42.170 ","End":"01:43.640","Text":"I\u0027m going to get this bit here."},{"Start":"01:43.640 ","End":"01:44.900","Text":"It looks exactly like this,"},{"Start":"01:44.900 ","End":"01:48.185","Text":"except that this t is missing in this t is missing because they\u0027re 1."},{"Start":"01:48.185 ","End":"01:49.810","Text":"We plug in 0,"},{"Start":"01:49.810 ","End":"01:51.660","Text":"e^0 is 1,"},{"Start":"01:51.660 ","End":"01:53.895","Text":"st is 0 plus 1 is 1,"},{"Start":"01:53.895 ","End":"01:56.445","Text":"so we\u0027re just left with minus 1 over s^2."},{"Start":"01:56.445 ","End":"01:58.520","Text":"Of course there\u0027s a subtraction here,"},{"Start":"01:58.520 ","End":"02:00.140","Text":"and this simplifies to this."},{"Start":"02:00.140 ","End":"02:01.790","Text":"If you go back and check,"},{"Start":"02:01.790 ","End":"02:04.685","Text":"this is what I told you it would be."},{"Start":"02:04.685 ","End":"02:06.350","Text":"That\u0027s one of the integrals."},{"Start":"02:06.350 ","End":"02:07.685","Text":"Now let\u0027s do the other one."},{"Start":"02:07.685 ","End":"02:11.191","Text":"The second one is actually an improper integral because of the infinity,"},{"Start":"02:11.191 ","End":"02:12.950","Text":"we have to be more careful."},{"Start":"02:12.950 ","End":"02:15.530","Text":"First step is to split it up into two,"},{"Start":"02:15.530 ","End":"02:17.180","Text":"this 2 minus t,"},{"Start":"02:17.180 ","End":"02:21.085","Text":"is 1 integral minus another integral and the 2 comes out in front."},{"Start":"02:21.085 ","End":"02:23.480","Text":"This bit, the e to the minus st,"},{"Start":"02:23.480 ","End":"02:28.400","Text":"it\u0027s just divided by minus s. The integral is with respect to t so we"},{"Start":"02:28.400 ","End":"02:33.680","Text":"just divide by this minus this to get minus 1 over s. This one,"},{"Start":"02:33.680 ","End":"02:37.325","Text":"well, we\u0027ve just saw that this e to the minus st times t,"},{"Start":"02:37.325 ","End":"02:38.870","Text":"and this was the answer."},{"Start":"02:38.870 ","End":"02:43.205","Text":"In both of these cases we have to plug in 1 and infinity,"},{"Start":"02:43.205 ","End":"02:46.790","Text":"but of course infinity is not really a number."},{"Start":"02:46.790 ","End":"02:49.535","Text":"We\u0027re talking about a limit as something goes to infinity."},{"Start":"02:49.535 ","End":"02:52.880","Text":"Let me restrict things to s bigger than 0 because I can"},{"Start":"02:52.880 ","End":"02:57.425","Text":"see that\u0027s going to converge this integral for f bigger than 0."},{"Start":"02:57.425 ","End":"03:00.950","Text":"Let\u0027s see, the first one is easier."},{"Start":"03:00.950 ","End":"03:06.860","Text":"You can almost treat infinity like a number because if s is positive,"},{"Start":"03:06.860 ","End":"03:09.230","Text":"then t goes to infinity,"},{"Start":"03:09.230 ","End":"03:11.705","Text":"minus st goes to minus infinity,"},{"Start":"03:11.705 ","End":"03:14.000","Text":"and e to the minus infinity is 0."},{"Start":"03:14.000 ","End":"03:18.805","Text":"It doesn\u0027t matter that is multiplied by 2 and minus 1/s, it\u0027s still 0."},{"Start":"03:18.805 ","End":"03:20.080","Text":"Plug in the 1."},{"Start":"03:20.080 ","End":"03:24.055","Text":"It looks very much like this except that the t is missing because it\u0027s 1."},{"Start":"03:24.055 ","End":"03:27.890","Text":"Here, there\u0027s a tricky part at the infinity."},{"Start":"03:27.890 ","End":"03:29.300","Text":"Let\u0027s just leave the infinity part."},{"Start":"03:29.300 ","End":"03:34.085","Text":"The one part is easy because just put t=1 here and here,"},{"Start":"03:34.085 ","End":"03:35.510","Text":"and this is what we get."},{"Start":"03:35.510 ","End":"03:38.500","Text":"The infinity part is actually a limit."},{"Start":"03:38.500 ","End":"03:40.070","Text":"If we multiply it out,"},{"Start":"03:40.070 ","End":"03:43.940","Text":"what we get is this times this."},{"Start":"03:43.940 ","End":"03:47.105","Text":"The minus minus makes it plus."},{"Start":"03:47.105 ","End":"03:48.890","Text":"The s/s^2."},{"Start":"03:48.890 ","End":"03:53.300","Text":"We get 1/s times e"},{"Start":"03:53.300 ","End":"03:56.360","Text":"to the minus st times"},{"Start":"03:56.360 ","End":"03:59.690","Text":"t. It doesn\u0027t matter if you take with the minus or without the minus,"},{"Start":"03:59.690 ","End":"04:01.745","Text":"because I\u0027m going to show that this goes to 0."},{"Start":"04:01.745 ","End":"04:09.880","Text":"But as it is, this part goes to 0 and this part goes to infinity."},{"Start":"04:09.880 ","End":"04:11.990","Text":"We have 0 times infinity case."},{"Start":"04:11.990 ","End":"04:13.550","Text":"Anyway, I\u0027ll do this limit at the end,"},{"Start":"04:13.550 ","End":"04:15.590","Text":"let\u0027s just continue here,"},{"Start":"04:15.590 ","End":"04:23.595","Text":"I just wrote it in red to remind me to do it at the end and need to simplify now."},{"Start":"04:23.595 ","End":"04:25.920","Text":"The first square bracket comes out,"},{"Start":"04:25.920 ","End":"04:28.475","Text":"minus minus is plus and the 2 goes on top."},{"Start":"04:28.475 ","End":"04:30.800","Text":"The second part, minus,"},{"Start":"04:30.800 ","End":"04:34.110","Text":"minus, minus is minus,"},{"Start":"04:34.110 ","End":"04:38.030","Text":"and we have here just this,"},{"Start":"04:38.030 ","End":"04:40.915","Text":"e to the minus s, s plus 1 over s^2,"},{"Start":"04:40.915 ","End":"04:42.795","Text":"this part was 0."},{"Start":"04:42.795 ","End":"04:44.945","Text":"Common denominator is s^2,"},{"Start":"04:44.945 ","End":"04:51.890","Text":"this part is multiplied by s and we subtract the numerators here, collect like terms."},{"Start":"04:51.890 ","End":"04:53.630","Text":"There\u0027s an se to the minus s here,"},{"Start":"04:53.630 ","End":"04:55.865","Text":"and here it\u0027s 2 minus 1 is 1,"},{"Start":"04:55.865 ","End":"05:00.560","Text":"and that\u0027s what we got as the answer for the second integral."},{"Start":"05:00.560 ","End":"05:04.230","Text":"The final debt is this 0."},{"Start":"05:04.230 ","End":"05:06.050","Text":"I\u0027ve to show you a limit,"},{"Start":"05:06.050 ","End":"05:09.845","Text":"need the limit of this thing as t goes to infinity."},{"Start":"05:09.845 ","End":"05:11.600","Text":"It didn\u0027t really need the 1/s,"},{"Start":"05:11.600 ","End":"05:14.270","Text":"it doesn\u0027t hurt anyway, it\u0027s a constant as far as t goes."},{"Start":"05:14.270 ","End":"05:17.515","Text":"What we do is because it\u0027s a 0 times infinity,"},{"Start":"05:17.515 ","End":"05:20.090","Text":"we put one of them in the denominator."},{"Start":"05:20.090 ","End":"05:24.260","Text":"I put the e to the minus st in the denominator and it becomes plus"},{"Start":"05:24.260 ","End":"05:29.660","Text":"st. Now instead of having a 0 times infinity,"},{"Start":"05:29.660 ","End":"05:32.765","Text":"we now have an infinity over infinity case."},{"Start":"05:32.765 ","End":"05:36.005","Text":"We can use L\u0027Hopital\u0027s rule,"},{"Start":"05:36.005 ","End":"05:39.845","Text":"differentiate the numerator, differentiate the denominator,"},{"Start":"05:39.845 ","End":"05:42.155","Text":"and then we get this over this."},{"Start":"05:42.155 ","End":"05:49.745","Text":"Now the limit is clear because e^st is e to the infinity, which is infinity."},{"Start":"05:49.745 ","End":"05:51.290","Text":"On the other bits are constants,"},{"Start":"05:51.290 ","End":"05:53.495","Text":"so 1 over infinity is 0,"},{"Start":"05:53.495 ","End":"05:55.535","Text":"and that\u0027s this 0 here."},{"Start":"05:55.535 ","End":"05:57.800","Text":"That settles all my debts,"},{"Start":"05:57.800 ","End":"06:00.540","Text":"and we are done."}],"ID":7908},{"Watched":false,"Name":"Exercise 15","Duration":"7m 28s","ChapterTopicVideoID":7857,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.375","Text":"In this exercise, we\u0027re going to compute the Laplace transform of a periodic function."},{"Start":"00:06.375 ","End":"00:08.400","Text":"We\u0027re not even given the function,"},{"Start":"00:08.400 ","End":"00:10.410","Text":"we\u0027re just given a sketch of it."},{"Start":"00:10.410 ","End":"00:13.215","Text":"So we have to find the formula."},{"Start":"00:13.215 ","End":"00:15.300","Text":"It\u0027s periodic as you can see,"},{"Start":"00:15.300 ","End":"00:19.350","Text":"that every 2 units it repeats itself."},{"Start":"00:19.350 ","End":"00:22.260","Text":"We didn\u0027t cover periodic functions in the tutorial,"},{"Start":"00:22.260 ","End":"00:23.759","Text":"there is a formula."},{"Start":"00:23.759 ","End":"00:26.490","Text":"The right time I\u0027ll produce that formula."},{"Start":"00:26.490 ","End":"00:31.115","Text":"The first thing we have to do is find an equation for this function,"},{"Start":"00:31.115 ","End":"00:34.190","Text":"or at least to find it between 0 and 2,"},{"Start":"00:34.190 ","End":"00:37.030","Text":"we just need to take 1 period of the function."},{"Start":"00:37.030 ","End":"00:38.720","Text":"Where the periodic function,"},{"Start":"00:38.720 ","End":"00:43.325","Text":"the period is usually called Omega as the little Greek letter Omega."},{"Start":"00:43.325 ","End":"00:46.205","Text":"In our case it\u0027s equal to 2."},{"Start":"00:46.205 ","End":"00:50.855","Text":"What we\u0027re going to do is write the equation of,"},{"Start":"00:50.855 ","End":"00:53.885","Text":"this will be g(t) from 0-2."},{"Start":"00:53.885 ","End":"00:55.700","Text":"We\u0027ll write it piecewise."},{"Start":"00:55.700 ","End":"00:59.405","Text":"There\u0027ll be this piece here that\u0027ll have one formula,"},{"Start":"00:59.405 ","End":"01:01.745","Text":"another formula for this bit here."},{"Start":"01:01.745 ","End":"01:03.590","Text":"We just needed from 0-2,"},{"Start":"01:03.590 ","End":"01:05.855","Text":"just 1 period of the function."},{"Start":"01:05.855 ","End":"01:08.750","Text":"We know all about linear functions."},{"Start":"01:08.750 ","End":"01:11.630","Text":"The first bit is 0-1,"},{"Start":"01:11.630 ","End":"01:14.165","Text":"and then later we\u0027ll do from 1-2."},{"Start":"01:14.165 ","End":"01:18.010","Text":"Now, linear function is y=at plus b."},{"Start":"01:18.010 ","End":"01:19.950","Text":"We\u0027re looking for a and b here."},{"Start":"01:19.950 ","End":"01:22.510","Text":"Since it goes through 0, 0,"},{"Start":"01:22.510 ","End":"01:27.815","Text":"we can substitute that in this equation and we get that b is 0."},{"Start":"01:27.815 ","End":"01:31.185","Text":"Next we\u0027re going to substitute 1,1."},{"Start":"01:31.185 ","End":"01:33.440","Text":"If we do that and do the computation,"},{"Start":"01:33.440 ","End":"01:35.585","Text":"we\u0027ll get that a is 1."},{"Start":"01:35.585 ","End":"01:38.765","Text":"That means that y is equal to,"},{"Start":"01:38.765 ","End":"01:42.110","Text":"we could write it as 1t plus 0,"},{"Start":"01:42.110 ","End":"01:46.580","Text":"but that\u0027s just equal to t. That\u0027s the first part."},{"Start":"01:46.580 ","End":"01:50.455","Text":"Now let\u0027s do the second bit from 1-2."},{"Start":"01:50.455 ","End":"01:54.425","Text":"Then we don\u0027t have the picture right in front of us this moment."},{"Start":"01:54.425 ","End":"01:56.660","Text":"Once again, it\u0027s a linear function,"},{"Start":"01:56.660 ","End":"01:58.220","Text":"at plus b,"},{"Start":"01:58.220 ","End":"01:59.990","Text":"and go back and look at the picture."},{"Start":"01:59.990 ","End":"02:02.720","Text":"It also passes through 1,1,"},{"Start":"02:02.720 ","End":"02:05.005","Text":"which gives us this equation."},{"Start":"02:05.005 ","End":"02:07.215","Text":"It passes through 2,0,"},{"Start":"02:07.215 ","End":"02:11.660","Text":"that gives us another equation in a and b. Subtract"},{"Start":"02:11.660 ","End":"02:16.220","Text":"this equation from this and you\u0027ll get that a is minus 1."},{"Start":"02:16.220 ","End":"02:18.215","Text":"If a is minus 1,"},{"Start":"02:18.215 ","End":"02:21.685","Text":"then bring this to the other side, b is 2,"},{"Start":"02:21.685 ","End":"02:27.240","Text":"and so this becomes y=1t minus 2,"},{"Start":"02:27.240 ","End":"02:29.220","Text":"or just t minus 2."},{"Start":"02:29.220 ","End":"02:31.130","Text":"Now we piece the 2 halves together."},{"Start":"02:31.130 ","End":"02:32.780","Text":"This was the first part of the function,"},{"Start":"02:32.780 ","End":"02:34.280","Text":"the one that went up,"},{"Start":"02:34.280 ","End":"02:37.880","Text":"the second part went down,"},{"Start":"02:37.880 ","End":"02:39.875","Text":"and we have g(t)."},{"Start":"02:39.875 ","End":"02:43.385","Text":"Now I\u0027m going to show you how we do the Laplace transform"},{"Start":"02:43.385 ","End":"02:47.285","Text":"of a periodic function g. In general,"},{"Start":"02:47.285 ","End":"02:49.690","Text":"if the function has period Omega,"},{"Start":"02:49.690 ","End":"02:53.180","Text":"this is the formula for the Laplace transform."},{"Start":"02:53.180 ","End":"02:57.290","Text":"It looks very much like the regular definition with 2 differences,"},{"Start":"02:57.290 ","End":"02:59.630","Text":"instead of infinity here we have an Omega,"},{"Start":"02:59.630 ","End":"03:01.805","Text":"and there\u0027s also this denominator."},{"Start":"03:01.805 ","End":"03:04.285","Text":"Now, in our case, Omega is 2,"},{"Start":"03:04.285 ","End":"03:06.450","Text":"so this becomes this."},{"Start":"03:06.450 ","End":"03:07.700","Text":"Here, we\u0027ve lost g,"},{"Start":"03:07.700 ","End":"03:10.610","Text":"I think I\u0027ll just remind you what it is. We\u0027ll write it again."},{"Start":"03:10.610 ","End":"03:12.680","Text":"It was a piecewise function."},{"Start":"03:12.680 ","End":"03:22.625","Text":"We had from 0-1 and we had from 1-2 and here it was t,"},{"Start":"03:22.625 ","End":"03:25.655","Text":"and here it was 2 minus t."},{"Start":"03:25.655 ","End":"03:29.495","Text":"Now we\u0027re going to break this integral up instead of from 0-2,"},{"Start":"03:29.495 ","End":"03:31.540","Text":"0-1, and 1-2."},{"Start":"03:31.540 ","End":"03:35.705","Text":"G(t) from 0-1 is t,"},{"Start":"03:35.705 ","End":"03:42.970","Text":"and from 1-2 is 2 minus t and now we have this computation to do."},{"Start":"03:42.970 ","End":"03:45.740","Text":"As usual I\u0027m just going to give you the answer to"},{"Start":"03:45.740 ","End":"03:49.535","Text":"the integrals and at the end I\u0027ll give you the details."},{"Start":"03:49.535 ","End":"03:54.110","Text":"The first part comes out to be this."},{"Start":"03:54.110 ","End":"03:56.230","Text":"I\u0027ll show you at the end."},{"Start":"03:56.230 ","End":"04:00.495","Text":"This part here is this."},{"Start":"04:00.495 ","End":"04:06.065","Text":"We\u0027ll just continue and at the end of this I\u0027ll do the integrals."},{"Start":"04:06.065 ","End":"04:12.260","Text":"This is really the answer and the rest of it is just algebraic simplification,"},{"Start":"04:12.260 ","End":"04:14.225","Text":"so let\u0027s do the bit of algebra now."},{"Start":"04:14.225 ","End":"04:20.210","Text":"The numerator of this dividing line has an s^2 and an s^2 and I combine the numerator."},{"Start":"04:20.210 ","End":"04:26.525","Text":"Next 2 things, bringing the s^2 down into the denominator here and collect terms here."},{"Start":"04:26.525 ","End":"04:28.579","Text":"This is what we get."},{"Start":"04:28.579 ","End":"04:31.775","Text":"There\u0027s still a lot more simplification we can do,"},{"Start":"04:31.775 ","End":"04:35.210","Text":"I can simplify the numerator if I use the formula,"},{"Start":"04:35.210 ","End":"04:38.600","Text":"the special binomial expansion,"},{"Start":"04:38.600 ","End":"04:44.550","Text":"(a minus b)^2 in algebra is a^2 minus 2ab plus b^2."},{"Start":"04:45.430 ","End":"04:53.205","Text":"If we take a to be 1 and b to be e^minus s,"},{"Start":"04:53.205 ","End":"04:55.740","Text":"then we can simplify this,"},{"Start":"04:55.740 ","End":"04:58.575","Text":"get 1 minus e^minus s^2,"},{"Start":"04:58.575 ","End":"05:01.875","Text":"that\u0027s the a minus b^2."},{"Start":"05:01.875 ","End":"05:06.995","Text":"The other thing I can do is use another algebraic formula"},{"Start":"05:06.995 ","End":"05:13.475","Text":"that a^2 minus b^2 is a minus b,"},{"Start":"05:13.475 ","End":"05:17.115","Text":"a plus b or the other way round, it doesn\u0027t matter."},{"Start":"05:17.115 ","End":"05:22.930","Text":"If I take a to be 1 and b to be e^minus s,"},{"Start":"05:22.930 ","End":"05:25.580","Text":"then this breaks up into this times this."},{"Start":"05:25.580 ","End":"05:27.230","Text":"Now look, there\u0027s a common factor,"},{"Start":"05:27.230 ","End":"05:29.555","Text":"1 minus e^s,"},{"Start":"05:29.555 ","End":"05:31.730","Text":"so if I cancel it here,"},{"Start":"05:31.730 ","End":"05:33.670","Text":"it\u0027ll cancel one of the factors."},{"Start":"05:33.670 ","End":"05:37.370","Text":"I\u0027ll just cross out the 2 and this is our final answer."},{"Start":"05:37.370 ","End":"05:39.395","Text":"That\u0027s about as simple as it can get."},{"Start":"05:39.395 ","End":"05:44.240","Text":"But we\u0027re not fully done because I still owe you 2 integrals."},{"Start":"05:44.240 ","End":"05:47.585","Text":"The first integral was this."},{"Start":"05:47.585 ","End":"05:50.390","Text":"We\u0027ve had this one several times before."},{"Start":"05:50.390 ","End":"05:53.060","Text":"You can do it again as integration by parts."},{"Start":"05:53.060 ","End":"05:54.860","Text":"I\u0027m just giving you the answer."},{"Start":"05:54.860 ","End":"05:56.375","Text":"This is the indefinite integral,"},{"Start":"05:56.375 ","End":"06:00.964","Text":"and now we have to substitute 0 and 1 to get the definite integral."},{"Start":"06:00.964 ","End":"06:07.290","Text":"What I\u0027m substituting is t not s. Then if t is 1, we get this."},{"Start":"06:07.290 ","End":"06:10.020","Text":"If t is Naught, we get this."},{"Start":"06:10.020 ","End":"06:13.190","Text":"This is what we get, and if you go back and check,"},{"Start":"06:13.190 ","End":"06:16.565","Text":"this is what I declared before."},{"Start":"06:16.565 ","End":"06:20.120","Text":"This is the one that I highlighted in this color."},{"Start":"06:20.120 ","End":"06:23.690","Text":"The second integral we had was this,"},{"Start":"06:23.690 ","End":"06:26.915","Text":"break it up the 2 minus t,"},{"Start":"06:26.915 ","End":"06:28.460","Text":"2 with this separately,"},{"Start":"06:28.460 ","End":"06:30.530","Text":"and t with this separately."},{"Start":"06:30.530 ","End":"06:32.675","Text":"This is what we have."},{"Start":"06:32.675 ","End":"06:34.270","Text":"The first integral,"},{"Start":"06:34.270 ","End":"06:36.740","Text":"remember s is a constant as far as t goes,"},{"Start":"06:36.740 ","End":"06:39.410","Text":"so we just have to divide this by minus s,"},{"Start":"06:39.410 ","End":"06:43.550","Text":"multiply by minus 1 over s. The second integral,"},{"Start":"06:43.550 ","End":"06:46.330","Text":"we just had it a moment ago, is this."},{"Start":"06:46.330 ","End":"06:51.470","Text":"We have to just plug in for each of them 1 and 2 and then subtract."},{"Start":"06:51.470 ","End":"06:53.630","Text":"I\u0027m not going to dwell on these computations."},{"Start":"06:53.630 ","End":"06:58.490","Text":"The first one I plug-in 2 and I plug-in 1, I get this."},{"Start":"06:58.490 ","End":"07:01.040","Text":"Also here, the 2 parts is this,"},{"Start":"07:01.040 ","End":"07:03.230","Text":"the one part is this."},{"Start":"07:03.230 ","End":"07:05.860","Text":"Now I\u0027ll just have to simplify,"},{"Start":"07:05.860 ","End":"07:08.375","Text":"first the common denominator over s^2,"},{"Start":"07:08.375 ","End":"07:11.915","Text":"which is multiplying this by s top and bottom."},{"Start":"07:11.915 ","End":"07:15.320","Text":"After simplifying, this is what we get."},{"Start":"07:15.320 ","End":"07:18.575","Text":"This is in fact what we had before."},{"Start":"07:18.575 ","End":"07:23.045","Text":"It\u0027s the one I highlighted in this color and check that it\u0027s assignments we had."},{"Start":"07:23.045 ","End":"07:28.860","Text":"That\u0027s my debt, these 2 integrals, and we\u0027re done."}],"ID":7909},{"Watched":false,"Name":"Exercise 16","Duration":"4m 25s","ChapterTopicVideoID":7858,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we want to compute"},{"Start":"00:02.310 ","End":"00:06.390","Text":"the Laplace transform of the following periodic function."},{"Start":"00:06.390 ","End":"00:08.070","Text":"You can see it\u0027s periodic."},{"Start":"00:08.070 ","End":"00:09.360","Text":"It repeating."},{"Start":"00:09.360 ","End":"00:11.610","Text":"It looks like the period is 2,"},{"Start":"00:11.610 ","End":"00:16.200","Text":"we take from 0-2 and then goes over and over again."},{"Start":"00:16.200 ","End":"00:17.460","Text":"Not the greatest sketch,"},{"Start":"00:17.460 ","End":"00:20.985","Text":"but I think we can identify what the function is."},{"Start":"00:20.985 ","End":"00:30.150","Text":"As I said, the period is 2 and if we take it from 0 up to 2,"},{"Start":"00:30.150 ","End":"00:34.290","Text":"then we can see that from 0-1,"},{"Start":"00:34.290 ","End":"00:36.458","Text":"it\u0027s equal to 1,"},{"Start":"00:36.458 ","End":"00:40.800","Text":"and from 1-2, it\u0027s minus 1."},{"Start":"00:40.800 ","End":"00:45.181","Text":"Here the picture is not quite right but anyway."},{"Start":"00:45.181 ","End":"00:50.180","Text":"Next we want the formula for periodic functions."},{"Start":"00:50.180 ","End":"00:51.530","Text":"You might\u0027ve seen it before."},{"Start":"00:51.530 ","End":"00:53.555","Text":"If not, I\u0027ll give it to you again."},{"Start":"00:53.555 ","End":"00:58.715","Text":"The Laplace transform of this g(t) is the integral."},{"Start":"00:58.715 ","End":"01:03.752","Text":"It\u0027s just like the definition of non-periodic with 2 differences, that at infinity,"},{"Start":"01:03.752 ","End":"01:05.230","Text":"we have Omega here,"},{"Start":"01:05.230 ","End":"01:07.070","Text":"which in our case is going to be 2,"},{"Start":"01:07.070 ","End":"01:09.950","Text":"and there\u0027s also this bit in the denominator."},{"Start":"01:09.950 ","End":"01:15.270","Text":"Now, we can\u0027t substitute this g(t) directly because it\u0027s defined piece wise,"},{"Start":"01:15.270 ","End":"01:17.350","Text":"so we break it up into 2 bits."},{"Start":"01:17.350 ","End":"01:18.880","Text":"First of all, of course,"},{"Start":"01:18.880 ","End":"01:20.675","Text":"the Omega is 2."},{"Start":"01:20.675 ","End":"01:26.058","Text":"Now we can split it up from 0-1 and from 1-2,"},{"Start":"01:26.058 ","End":"01:28.410","Text":"just get some space here."},{"Start":"01:28.410 ","End":"01:31.230","Text":"Recall that from 0-1,"},{"Start":"01:31.230 ","End":"01:33.045","Text":"g(t) is 1,"},{"Start":"01:33.045 ","End":"01:34.710","Text":"and from 1-2,"},{"Start":"01:34.710 ","End":"01:38.265","Text":"g(t) is minus 1."},{"Start":"01:38.265 ","End":"01:41.120","Text":"To interrupt the flow,"},{"Start":"01:41.120 ","End":"01:44.600","Text":"I\u0027ll give you the solution to the integrals."},{"Start":"01:44.600 ","End":"01:47.435","Text":"This integral here is this,"},{"Start":"01:47.435 ","End":"01:50.510","Text":"and this integral here is this."},{"Start":"01:50.510 ","End":"01:53.600","Text":"I\u0027ll do the computation at the end."},{"Start":"01:53.600 ","End":"01:56.390","Text":"Next, we want to simplify this expression."},{"Start":"01:56.390 ","End":"01:58.070","Text":"I just cleared some space."},{"Start":"01:58.070 ","End":"01:59.690","Text":"We work on the numerator,"},{"Start":"01:59.690 ","End":"02:01.565","Text":"it has a common denominator,"},{"Start":"02:01.565 ","End":"02:06.950","Text":"s. It\u0027s this minus this and then we have another 1 of these minuses,"},{"Start":"02:06.950 ","End":"02:09.625","Text":"so that makes it twice and then this."},{"Start":"02:09.625 ","End":"02:14.150","Text":"Next to the 2 things, put the s in the denominator and identify this."},{"Start":"02:14.150 ","End":"02:19.350","Text":"It\u0027s in the form a^2 minus 2ab plus b^2,"},{"Start":"02:19.350 ","End":"02:23.100","Text":"which is a minus b^2."},{"Start":"02:23.100 ","End":"02:28.608","Text":"We take as a would be 1 and b is e^minus"},{"Start":"02:28.608 ","End":"02:35.360","Text":"s. This would be (1 minus e^minus s)^2 here."},{"Start":"02:35.360 ","End":"02:40.160","Text":"But I also reverse the order because if you reverse the order,"},{"Start":"02:40.160 ","End":"02:41.180","Text":"it makes it negative,"},{"Start":"02:41.180 ","End":"02:43.475","Text":"but if it\u0027s squared, it doesn\u0027t matter."},{"Start":"02:43.475 ","End":"02:45.440","Text":"Now we\u0027re up to here."},{"Start":"02:45.440 ","End":"02:48.530","Text":"Then I\u0027m going to decompose the denominator using"},{"Start":"02:48.530 ","End":"02:54.360","Text":"another algebraic rule that a^2 minus b^2 is a minus b,"},{"Start":"02:54.360 ","End":"02:58.444","Text":"a plus b, also known as the difference of squares formula."},{"Start":"02:58.444 ","End":"03:01.835","Text":"This here decomposes into this times this."},{"Start":"03:01.835 ","End":"03:04.075","Text":"Now look, we have a common factor,"},{"Start":"03:04.075 ","End":"03:10.620","Text":"1 minus e^minus s. I can cancel this with this,"},{"Start":"03:10.620 ","End":"03:13.940","Text":"and this is the answer to the exercise."},{"Start":"03:13.940 ","End":"03:17.900","Text":"If you want to see how I do those integrals from before, then stay,"},{"Start":"03:17.900 ","End":"03:23.360","Text":"otherwise you\u0027re free to leave. The integrals."},{"Start":"03:23.360 ","End":"03:25.145","Text":"The first one was this."},{"Start":"03:25.145 ","End":"03:27.005","Text":"And the integral is dt."},{"Start":"03:27.005 ","End":"03:28.660","Text":"This is straightforward."},{"Start":"03:28.660 ","End":"03:30.255","Text":"s is a constant,"},{"Start":"03:30.255 ","End":"03:34.490","Text":"so it\u0027s minus s. We just divide by that minus s. This is what we get."},{"Start":"03:34.490 ","End":"03:38.165","Text":"Next, we plug in 0 and 1 and subtract."},{"Start":"03:38.165 ","End":"03:43.150","Text":"We have this for 1 and this bit for 0."},{"Start":"03:43.150 ","End":"03:45.650","Text":"Minus minus is plus,"},{"Start":"03:45.650 ","End":"03:48.185","Text":"change the order, common denominator."},{"Start":"03:48.185 ","End":"03:51.290","Text":"This was the first of the integrals."},{"Start":"03:51.290 ","End":"03:52.430","Text":"You can go back and check it."},{"Start":"03:52.430 ","End":"03:54.035","Text":"This is what we had."},{"Start":"03:54.035 ","End":"03:56.270","Text":"Now let\u0027s go to the second one."},{"Start":"03:56.270 ","End":"03:58.805","Text":"Very similar to the previous one."},{"Start":"03:58.805 ","End":"04:00.680","Text":"It\u0027s the same e^minus st."},{"Start":"04:00.680 ","End":"04:04.644","Text":"It\u0027s got a minus here and the different limits."},{"Start":"04:04.644 ","End":"04:09.680","Text":"The integral before was with a minus and now we don\u0027t have the minus."},{"Start":"04:09.680 ","End":"04:12.605","Text":"We have to plug in 2 and 1 and subtract."},{"Start":"04:12.605 ","End":"04:14.315","Text":"Here\u0027s the 2,"},{"Start":"04:14.315 ","End":"04:16.250","Text":"and here\u0027s the 1,"},{"Start":"04:16.250 ","End":"04:19.273","Text":"let\u0027s throw out the brackets,"},{"Start":"04:19.273 ","End":"04:20.960","Text":"and the common denominator,"},{"Start":"04:20.960 ","End":"04:24.750","Text":"and this is what we had for our second integral. You can check. We\u0027re done."}],"ID":7910},{"Watched":false,"Name":"Exercise 17","Duration":"3m 12s","ChapterTopicVideoID":7859,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"In this exercise, we have to compute"},{"Start":"00:02.100 ","End":"00:05.459","Text":"the Laplace transform of the following periodic function,"},{"Start":"00:05.459 ","End":"00:09.060","Text":"which means that it\u0027s like this bit and then another one of those,"},{"Start":"00:09.060 ","End":"00:13.514","Text":"and it keeps on going at infinite item, or each period,"},{"Start":"00:13.514 ","End":"00:20.475","Text":"like the period from 0-2 is made up of this part and this part."},{"Start":"00:20.475 ","End":"00:27.480","Text":"Then it repeats. All we really need to get is the formula from 0-2,"},{"Start":"00:27.480 ","End":"00:33.310","Text":"and it\u0027s pretty clear that this bit has the formula where g(t)=t."},{"Start":"00:34.390 ","End":"00:39.125","Text":"This bit is where g(t) is the constant 1."},{"Start":"00:39.125 ","End":"00:40.880","Text":"This is from 0-1,"},{"Start":"00:40.880 ","End":"00:42.685","Text":"and this is from 1-2."},{"Start":"00:42.685 ","End":"00:47.960","Text":"Now we just need the formula for a periodic function."},{"Start":"00:47.960 ","End":"00:50.570","Text":"The Laplace transform of, I mean,"},{"Start":"00:50.570 ","End":"00:53.900","Text":"the Laplace transform of a periodic function is"},{"Start":"00:53.900 ","End":"00:57.890","Text":"given by this formula where Omega is the period,"},{"Start":"00:57.890 ","End":"01:01.300","Text":"and in our case, Omega is 2,"},{"Start":"01:01.300 ","End":"01:05.870","Text":"so just replace Omega by 2 here and here."},{"Start":"01:05.870 ","End":"01:09.740","Text":"This is the integral. We can substitute g(t) as is,"},{"Start":"01:09.740 ","End":"01:11.600","Text":"because it\u0027s defined in 2 different ways,"},{"Start":"01:11.600 ","End":"01:13.040","Text":"so we break this up."},{"Start":"01:13.040 ","End":"01:18.825","Text":"That\u0027s a standard method from 0-1 and from 1-2 separately."},{"Start":"01:18.825 ","End":"01:21.150","Text":"Here\u0027s the t here,"},{"Start":"01:21.150 ","End":"01:23.355","Text":"that\u0027s from 0-1,"},{"Start":"01:23.355 ","End":"01:25.920","Text":"and this bit one is here,"},{"Start":"01:25.920 ","End":"01:28.230","Text":"and that\u0027s what applies from 1-2."},{"Start":"01:28.230 ","End":"01:34.165","Text":"Now, we just have to evaluate the integrals and simplify."},{"Start":"01:34.165 ","End":"01:38.025","Text":"As usual, I\u0027m going to tell you what the integrals are."},{"Start":"01:38.025 ","End":"01:39.240","Text":"It\u0027s not to stop the flow."},{"Start":"01:39.240 ","End":"01:41.865","Text":"The first integral is this bit here,"},{"Start":"01:41.865 ","End":"01:45.620","Text":"and the second integral is this bit here."},{"Start":"01:45.620 ","End":"01:47.480","Text":"Essentially, this is the answer,"},{"Start":"01:47.480 ","End":"01:49.220","Text":"but we want to simplify it."},{"Start":"01:49.220 ","End":"01:53.075","Text":"Put the s^2 into the denominator,"},{"Start":"01:53.075 ","End":"02:00.540","Text":"but we also have to multiply this numerator by s and this by s to make it s^2."},{"Start":"02:00.540 ","End":"02:04.860","Text":"This as is and this multiplied by s gives us this."},{"Start":"02:04.860 ","End":"02:07.215","Text":"Just open the brackets,"},{"Start":"02:07.215 ","End":"02:09.990","Text":"and just collect like terms."},{"Start":"02:09.990 ","End":"02:12.140","Text":"This is the answer,"},{"Start":"02:12.140 ","End":"02:15.680","Text":"except that I have to show you how I did those integrals."},{"Start":"02:15.680 ","End":"02:18.130","Text":"The first integral is this."},{"Start":"02:18.130 ","End":"02:22.490","Text":"We\u0027ve seen this e^-st times t many times before."},{"Start":"02:22.490 ","End":"02:25.085","Text":"It\u0027s integral is this,"},{"Start":"02:25.085 ","End":"02:26.420","Text":"but it\u0027s a definite integral."},{"Start":"02:26.420 ","End":"02:29.755","Text":"We have to substitute 1 and then 0 and subtract."},{"Start":"02:29.755 ","End":"02:33.510","Text":"Plug in 1 and get this, plug in 0,"},{"Start":"02:33.510 ","End":"02:35.280","Text":"and we get this,"},{"Start":"02:35.280 ","End":"02:38.360","Text":"change the order, common denominator."},{"Start":"02:38.360 ","End":"02:41.330","Text":"This is the answer for the first integral and check it."},{"Start":"02:41.330 ","End":"02:42.919","Text":"This is what we had before."},{"Start":"02:42.919 ","End":"02:44.750","Text":"Now, let\u0027s go on to the second."},{"Start":"02:44.750 ","End":"02:46.460","Text":"That was this integral."},{"Start":"02:46.460 ","End":"02:48.785","Text":"This is even easier."},{"Start":"02:48.785 ","End":"02:50.300","Text":"We\u0027ve seen this before too."},{"Start":"02:50.300 ","End":"02:52.040","Text":"We just divide by minus s,"},{"Start":"02:52.040 ","End":"02:54.740","Text":"or multiply by minus 1 over s. Now,"},{"Start":"02:54.740 ","End":"02:56.330","Text":"we have to substitute,"},{"Start":"02:56.330 ","End":"02:58.115","Text":"plug in 2,"},{"Start":"02:58.115 ","End":"03:00.865","Text":"then plug in 1 and subtract."},{"Start":"03:00.865 ","End":"03:03.710","Text":"Change the order, put a common denominator."},{"Start":"03:03.710 ","End":"03:10.205","Text":"This is what we get and you should check that this is what we had before."},{"Start":"03:10.205 ","End":"03:12.990","Text":"We are done."}],"ID":7911},{"Watched":false,"Name":"Exercise 18","Duration":"3m 14s","ChapterTopicVideoID":7860,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.080","Text":"This exercise is not really an exercise,"},{"Start":"00:03.080 ","End":"00:06.599","Text":"it is a brief tutorial on the step function."},{"Start":"00:06.599 ","End":"00:09.555","Text":"The step function is a discontinuous function."},{"Start":"00:09.555 ","End":"00:13.140","Text":"It\u0027s 0 for all negative values of t,"},{"Start":"00:13.140 ","End":"00:19.200","Text":"and then it jumps to 1 for non-negative values of t. The reason we use"},{"Start":"00:19.200 ","End":"00:25.305","Text":"the letter u is its full name is the unit step function."},{"Start":"00:25.305 ","End":"00:27.330","Text":"And it also has other letters,"},{"Start":"00:27.330 ","End":"00:30.255","Text":"most commonly big H,"},{"Start":"00:30.255 ","End":"00:35.054","Text":"after the mathematician, Oliver Heaviside."},{"Start":"00:35.054 ","End":"00:37.650","Text":"He was also electrical engineer and physicist."},{"Start":"00:37.650 ","End":"00:41.745","Text":"It\u0027s sometimes called the Heaviside step function."},{"Start":"00:41.745 ","End":"00:45.080","Text":"I\u0027m just letting you know in case you see this letter,"},{"Start":"00:45.080 ","End":"00:47.870","Text":"I think there\u0027s even other letters that they use capital Theta."},{"Start":"00:47.870 ","End":"00:50.555","Text":"Anyway, we\u0027re going to stick to a little u."},{"Start":"00:50.555 ","End":"00:52.745","Text":"Although it\u0027s the unit\u0027s step function,"},{"Start":"00:52.745 ","End":"00:55.555","Text":"usually we\u0027ll just say step function."},{"Start":"00:55.555 ","End":"00:57.380","Text":"Here\u0027s what it looks like."},{"Start":"00:57.380 ","End":"00:59.360","Text":"This is 0, the origin,"},{"Start":"00:59.360 ","End":"01:02.930","Text":"the y-axis, and the x-axis."},{"Start":"01:02.930 ","End":"01:04.530","Text":"Well, really this is t,"},{"Start":"01:04.530 ","End":"01:06.670","Text":"I\u0027ll just label this u(t)."},{"Start":"01:06.670 ","End":"01:10.140","Text":"It\u0027s 0 for all the negative values,"},{"Start":"01:10.140 ","End":"01:14.145","Text":"and then it makes a jump and becomes 1."},{"Start":"01:14.145 ","End":"01:16.640","Text":"I could have put a hollow circle here,"},{"Start":"01:16.640 ","End":"01:18.770","Text":"but the solid circle here means that,"},{"Start":"01:18.770 ","End":"01:21.110","Text":"that 0 itself it\u0027s 1."},{"Start":"01:21.110 ","End":"01:27.890","Text":"Now the next thing we want to say is that sometimes we want this function shifted."},{"Start":"01:27.890 ","End":"01:30.139","Text":"If I don\u0027t want it to jump at 0,"},{"Start":"01:30.139 ","End":"01:36.455","Text":"let\u0027s say I wanted it to jump at t=3 or t=2 or t equals minus 4,"},{"Start":"01:36.455 ","End":"01:38.740","Text":"I can modify this slightly,"},{"Start":"01:38.740 ","End":"01:41.930","Text":"and in general, we know how to shift a function."},{"Start":"01:41.930 ","End":"01:44.620","Text":"Suppose I wanted to shift it,"},{"Start":"01:44.620 ","End":"01:45.650","Text":"to make it general,"},{"Start":"01:45.650 ","End":"01:49.520","Text":"I want to move it k units to the right."},{"Start":"01:49.520 ","End":"01:55.760","Text":"Then in that case we replace t by t minus k. In fact here\u0027s the picture of what I mean."},{"Start":"01:55.760 ","End":"01:58.880","Text":"We just shifted k units to the right,"},{"Start":"01:58.880 ","End":"02:01.430","Text":"and the formula, like I said,"},{"Start":"02:01.430 ","End":"02:04.935","Text":"we replace t with t minus k,"},{"Start":"02:04.935 ","End":"02:07.330","Text":"so we have u(t minus k)."},{"Start":"02:07.330 ","End":"02:08.860","Text":"If I go to the definition,"},{"Start":"02:08.860 ","End":"02:16.565","Text":"that will be 0 if t minus k is negative and 1 if t minus k is non-negative."},{"Start":"02:16.565 ","End":"02:20.090","Text":"But I can slightly rewrite this as follows,"},{"Start":"02:20.090 ","End":"02:22.490","Text":"that it\u0027s 0 when t is less than k,"},{"Start":"02:22.490 ","End":"02:24.695","Text":"I just bring the k to the other side and here is 1,"},{"Start":"02:24.695 ","End":"02:27.650","Text":"t bigger or equal to k. This is more intuitive."},{"Start":"02:27.650 ","End":"02:31.115","Text":"This means that we 0 up to k,"},{"Start":"02:31.115 ","End":"02:35.795","Text":"not including, and then at k we jump to the value 1."},{"Start":"02:35.795 ","End":"02:38.750","Text":"It\u0027s definitely a discontinuous function,"},{"Start":"02:38.750 ","End":"02:40.645","Text":"at least this point."},{"Start":"02:40.645 ","End":"02:42.465","Text":"In the books,"},{"Start":"02:42.465 ","End":"02:46.655","Text":"some professors will use this notation instead of this."},{"Start":"02:46.655 ","End":"02:54.530","Text":"They\u0027ll say u_k(t) to indicate the unit step function which jumps at k,"},{"Start":"02:54.530 ","End":"02:55.775","Text":"so if our k was,"},{"Start":"02:55.775 ","End":"02:57.050","Text":"let\u0027s say 2,"},{"Start":"02:57.050 ","End":"03:01.850","Text":"then this function could be written as u(t) minus 2,"},{"Start":"03:01.850 ","End":"03:06.425","Text":"but it could also be written as u_2(t)."},{"Start":"03:06.425 ","End":"03:08.415","Text":"All these are equivalent."},{"Start":"03:08.415 ","End":"03:10.310","Text":"We mostly will not be using this,"},{"Start":"03:10.310 ","End":"03:12.020","Text":"I\u0027ll just be using this."},{"Start":"03:12.020 ","End":"03:14.760","Text":"That\u0027s it for the intro."}],"ID":7912},{"Watched":false,"Name":"Exercise 19","Duration":"2m 51s","ChapterTopicVideoID":7861,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"In this exercise, we have to sketch the function f(t),"},{"Start":"00:03.975 ","End":"00:06.945","Text":"which is u(t) minus 2,"},{"Start":"00:06.945 ","End":"00:08.910","Text":"minus u(t) minus 3,"},{"Start":"00:08.910 ","End":"00:12.060","Text":"where u(t) is the unit step function,"},{"Start":"00:12.060 ","End":"00:13.545","Text":"I don\u0027t always say unit."},{"Start":"00:13.545 ","End":"00:14.988","Text":"Let\u0027s see. Now,"},{"Start":"00:14.988 ","End":"00:16.140","Text":"it\u0027s made up of 2 pieces,"},{"Start":"00:16.140 ","End":"00:19.155","Text":"each of them is u(t) minus something."},{"Start":"00:19.155 ","End":"00:22.785","Text":"In the previous clip, we talked about u(t) minus something,"},{"Start":"00:22.785 ","End":"00:25.830","Text":"t minus k, that is the shifted unit function,"},{"Start":"00:25.830 ","End":"00:27.945","Text":"and it\u0027s shifted k units to the right,"},{"Start":"00:27.945 ","End":"00:31.200","Text":"which means that it\u0027s 0 when t is less than k,"},{"Start":"00:31.200 ","End":"00:35.420","Text":"and 1 when t is greater or equal to k. For the first piece,"},{"Start":"00:35.420 ","End":"00:36.955","Text":"we have k=2,"},{"Start":"00:36.955 ","End":"00:44.066","Text":"so u(t) minus 2 is 0 or 1 depending on where t is in relation to 2,"},{"Start":"00:44.066 ","End":"00:46.250","Text":"if it\u0027s less than, or greater than, or equal to."},{"Start":"00:46.250 ","End":"00:51.060","Text":"Similarly, u(t) minus 3 is this 0,"},{"Start":"00:51.060 ","End":"00:52.148","Text":"and we\u0027re less than 3,"},{"Start":"00:52.148 ","End":"00:55.100","Text":"and from 3 onwards we\u0027re equal to 1."},{"Start":"00:55.100 ","End":"00:58.235","Text":"So now, we want to do a subtraction,"},{"Start":"00:58.235 ","End":"01:05.420","Text":"but we have to divide up into cases because things change every time we cross the 2,"},{"Start":"01:05.420 ","End":"01:07.550","Text":"something changes, and when we cross the 3,"},{"Start":"01:07.550 ","End":"01:09.229","Text":"also something changes,"},{"Start":"01:09.229 ","End":"01:14.045","Text":"so we\u0027re going to have to write this as a piecewise function in 3 pieces."},{"Start":"01:14.045 ","End":"01:16.270","Text":"Well, let me just slow down a bit."},{"Start":"01:16.270 ","End":"01:20.300","Text":"If I was just to try and do it naively and subtract this minus this,"},{"Start":"01:20.300 ","End":"01:22.525","Text":"I would say it\u0027s this minus this,"},{"Start":"01:22.525 ","End":"01:26.555","Text":"and let me just copy this and this and put a minus sign and say it\u0027s this minus this."},{"Start":"01:26.555 ","End":"01:30.050","Text":"But how do we subtract such piecewise things?"},{"Start":"01:30.050 ","End":"01:35.990","Text":"Well, we notice that there are cases that depending on where we are relative to 2 and 3,"},{"Start":"01:35.990 ","End":"01:37.685","Text":"we get different things."},{"Start":"01:37.685 ","End":"01:42.125","Text":"Then we get the idea that we\u0027d better split it up into t less than 2,"},{"Start":"01:42.125 ","End":"01:43.985","Text":"between 2 and 3,"},{"Start":"01:43.985 ","End":"01:46.205","Text":"and greater than 3."},{"Start":"01:46.205 ","End":"01:49.594","Text":"Now, when t is less than 2,"},{"Start":"01:49.594 ","End":"01:52.295","Text":"then t is also less than 3,"},{"Start":"01:52.295 ","End":"01:56.810","Text":"so we get from here 0 and from here 0,"},{"Start":"01:56.810 ","End":"01:59.090","Text":"and that gives me 0."},{"Start":"01:59.090 ","End":"02:02.240","Text":"When t is between 2 and 3,"},{"Start":"02:02.240 ","End":"02:05.600","Text":"bigger or equal to 2 and less than 3 strictly,"},{"Start":"02:05.600 ","End":"02:07.625","Text":"then from here we get the 1,"},{"Start":"02:07.625 ","End":"02:10.320","Text":"and from here, we\u0027re less than 3 still,"},{"Start":"02:10.320 ","End":"02:13.055","Text":"so it\u0027s 0, and that\u0027s equal to 1."},{"Start":"02:13.055 ","End":"02:14.230","Text":"That\u0027s the 1 here."},{"Start":"02:14.230 ","End":"02:16.505","Text":"When t is bigger or equal to 3,"},{"Start":"02:16.505 ","End":"02:18.790","Text":"it\u0027s certainly also bigger or equal to 2,"},{"Start":"02:18.790 ","End":"02:22.010","Text":"and so we get 1 minus 1,"},{"Start":"02:22.010 ","End":"02:23.600","Text":"which is 0 again."},{"Start":"02:23.600 ","End":"02:25.213","Text":"Everywhere we\u0027re 0,"},{"Start":"02:25.213 ","End":"02:30.065","Text":"except when 2 less than or equal to t less than 3."},{"Start":"02:30.065 ","End":"02:33.230","Text":"Now, if I produce a sketch of that,"},{"Start":"02:33.230 ","End":"02:35.486","Text":"we can see up to 2,"},{"Start":"02:35.486 ","End":"02:38.470","Text":"not including 2,0, that\u0027s this bit."},{"Start":"02:38.470 ","End":"02:40.939","Text":"Then when we get to 2,"},{"Start":"02:40.939 ","End":"02:42.980","Text":"we become 1,"},{"Start":"02:42.980 ","End":"02:45.788","Text":"and that goes on as long as we\u0027re less than 3,"},{"Start":"02:45.788 ","End":"02:49.190","Text":"and then from 3 onwards, 0 again."},{"Start":"02:49.190 ","End":"02:52.570","Text":"This is what it looks like, and we\u0027re done."}],"ID":7913},{"Watched":false,"Name":"Exercise 20","Duration":"1m 20s","ChapterTopicVideoID":7862,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we\u0027re given the following function,"},{"Start":"00:02.760 ","End":"00:04.200","Text":"0 and t is less than 4,"},{"Start":"00:04.200 ","End":"00:06.165","Text":"1 when t is bigger or equal to 4,"},{"Start":"00:06.165 ","End":"00:08.910","Text":"we want to express this in terms of the step function."},{"Start":"00:08.910 ","End":"00:10.890","Text":"Now, a couple of clips ago,"},{"Start":"00:10.890 ","End":"00:15.300","Text":"we defined the unit step function and some variants of"},{"Start":"00:15.300 ","End":"00:20.130","Text":"it and if you were paying attention and remembered at all,"},{"Start":"00:20.130 ","End":"00:24.285","Text":"you would be able to write the answers straightaway as u(t minus 4)."},{"Start":"00:24.285 ","End":"00:27.510","Text":"But I\u0027m going to take it more slowly."},{"Start":"00:27.510 ","End":"00:29.640","Text":"I don\u0027t do a bit of a review too."},{"Start":"00:29.640 ","End":"00:34.214","Text":"Recall, we defined the unit step function as follows,"},{"Start":"00:34.214 ","End":"00:36.000","Text":"0 when t is negative,"},{"Start":"00:36.000 ","End":"00:38.205","Text":"1 when t is bigger or equal to 0."},{"Start":"00:38.205 ","End":"00:40.305","Text":"Then we said that if we want to shift it,"},{"Start":"00:40.305 ","End":"00:41.525","Text":"k units to the right,"},{"Start":"00:41.525 ","End":"00:48.750","Text":"then the result is u(t minus k) which turns out to be 0 for t less than k,"},{"Start":"00:48.750 ","End":"00:51.170","Text":"1t greater, or equal to k. Of course,"},{"Start":"00:51.170 ","End":"00:54.725","Text":"if we shift to the left this thing becomes positive."},{"Start":"00:54.725 ","End":"00:57.620","Text":"If we wanted to shift it left 3,"},{"Start":"00:57.620 ","End":"01:00.950","Text":"then it would be t plus 3."},{"Start":"01:00.950 ","End":"01:02.675","Text":"Now if we look at this and we look at this,"},{"Start":"01:02.675 ","End":"01:07.390","Text":"isn\u0027t this exactly the same except with k=4?"},{"Start":"01:07.390 ","End":"01:10.585","Text":"Just say it\u0027s the same thing with k=4."},{"Start":"01:10.585 ","End":"01:14.510","Text":"If that\u0027s the case, then I can say what it equals to by putting"},{"Start":"01:14.510 ","End":"01:20.580","Text":"k=4 and so the answer is u(t minus 4). We\u0027re done."}],"ID":7914},{"Watched":false,"Name":"Exercise 21","Duration":"3m 2s","ChapterTopicVideoID":7863,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.210","Text":"In this exercise, we\u0027re continuing with the unit step function and we\u0027re going to discuss"},{"Start":"00:06.210 ","End":"00:09.330","Text":"its Laplace transform and also the transform of"},{"Start":"00:09.330 ","End":"00:14.670","Text":"some variations of this step function or combining it with another function."},{"Start":"00:14.670 ","End":"00:16.005","Text":"There\u0027s 3 parts,"},{"Start":"00:16.005 ","End":"00:20.159","Text":"first one is just the transform of the unit step function itself."},{"Start":"00:20.159 ","End":"00:23.730","Text":"In b, we want it shifted to the right."},{"Start":"00:23.730 ","End":"00:29.504","Text":"In c, we want to take this and multiply it by some function"},{"Start":"00:29.504 ","End":"00:36.570","Text":"f(t) minus k. Assuming that we have the Laplace transform of f. In the first part,"},{"Start":"00:36.570 ","End":"00:40.820","Text":"I\u0027m just reminding you what the unit step function is defined this way and it"},{"Start":"00:40.820 ","End":"00:45.960","Text":"actually appears in the table of Laplace transforms."},{"Start":"00:45.960 ","End":"00:49.145","Text":"The answer is just 1 over s from the table."},{"Start":"00:49.145 ","End":"00:51.875","Text":"Now, let\u0027s continue to part b."},{"Start":"00:51.875 ","End":"00:56.000","Text":"In part b, we have the shifted unit step function,"},{"Start":"00:56.000 ","End":"00:57.635","Text":"shifted k to the right,"},{"Start":"00:57.635 ","End":"01:00.170","Text":"which means that the jump occurs at k,"},{"Start":"01:00.170 ","End":"01:04.910","Text":"and this is also called u_k(t),"},{"Start":"01:04.910 ","End":"01:12.515","Text":"and this also appears in the table of Laplace transforms and this is the answer."},{"Start":"01:12.515 ","End":"01:15.320","Text":"Just note that if you put k=0 here,"},{"Start":"01:15.320 ","End":"01:16.670","Text":"e^0 is 1,"},{"Start":"01:16.670 ","End":"01:20.510","Text":"so we get the same as the previous and that\u0027s nice to know."},{"Start":"01:20.510 ","End":"01:22.780","Text":"Now, the last one,"},{"Start":"01:22.780 ","End":"01:26.614","Text":"we have some function f and then it\u0027s quite common"},{"Start":"01:26.614 ","End":"01:30.785","Text":"in engineering to consider this expression."},{"Start":"01:30.785 ","End":"01:35.360","Text":"What it actually means is that we take a function f,"},{"Start":"01:35.360 ","End":"01:37.445","Text":"I\u0027m going to describe it geometrically."},{"Start":"01:37.445 ","End":"01:40.835","Text":"We shift f k units to the right,"},{"Start":"01:40.835 ","End":"01:45.440","Text":"but then we truncate it to the left of k. To the left of the value k,"},{"Start":"01:45.440 ","End":"01:48.935","Text":"it\u0027s all 0\u0027s, so it\u0027s a shift plus a truncation."},{"Start":"01:48.935 ","End":"01:50.675","Text":"I don\u0027t know if I should sketch it."},{"Start":"01:50.675 ","End":"01:53.950","Text":"I\u0027ll sketch it in a moment, I\u0027ll just give you the result first."},{"Start":"01:53.950 ","End":"01:56.475","Text":"This is what is written here."},{"Start":"01:56.475 ","End":"02:00.020","Text":"What it means is that if we know the Laplace transform of f,"},{"Start":"02:00.020 ","End":"02:01.820","Text":"and then we do this to f,"},{"Start":"02:01.820 ","End":"02:06.650","Text":"it has the effect of multiplying the transform by e to the minus ks."},{"Start":"02:06.650 ","End":"02:09.910","Text":"But basically it\u0027s done unless you want to see my rough sketch."},{"Start":"02:09.910 ","End":"02:17.494","Text":"Let\u0027s say I have some axes and then the function f might be something like this."},{"Start":"02:17.494 ","End":"02:23.900","Text":"This is 0, and let\u0027s say somewhere we have the value k and this is f,"},{"Start":"02:23.900 ","End":"02:27.890","Text":"this is t. Start by making a copy of"},{"Start":"02:27.890 ","End":"02:32.400","Text":"it with copy paste and I also want to mark this point on."},{"Start":"02:32.400 ","End":"02:37.280","Text":"Then I shifted to the right so that this point is above k and then I"},{"Start":"02:37.280 ","End":"02:43.165","Text":"truncate it like so and then this function is this."},{"Start":"02:43.165 ","End":"02:46.590","Text":"To the left of k, it\u0027s just all 0."},{"Start":"02:46.590 ","End":"02:51.500","Text":"It\u0027s 0 up to k, then f(t) minus k when we\u0027re"},{"Start":"02:51.500 ","End":"02:56.762","Text":"big or equal to k. If you needed a visualization but you don\u0027t and as I said,"},{"Start":"02:56.762 ","End":"03:02.250","Text":"this is the answer from the table of Fourier transforms. We\u0027re done."}],"ID":7915},{"Watched":false,"Name":"Exercise 22","Duration":"2m 46s","ChapterTopicVideoID":7864,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.475","Text":"In this exercise, we want to compute the Laplace transform of this function,"},{"Start":"00:05.475 ","End":"00:07.424","Text":"which is defined piecewise,"},{"Start":"00:07.424 ","End":"00:11.085","Text":"and we\u0027re going to do it with the help of the step function."},{"Start":"00:11.085 ","End":"00:15.970","Text":"Notice that we can take t minus 4 squared"},{"Start":"00:15.970 ","End":"00:21.645","Text":"outside the brackets because t minus 4 squared times 0 is still 0."},{"Start":"00:21.645 ","End":"00:25.700","Text":"This is going to help us because we recognize this bit in"},{"Start":"00:25.700 ","End":"00:30.245","Text":"the curly braces as a shift of the step function."},{"Start":"00:30.245 ","End":"00:36.120","Text":"In fact, this is just u(t) minus 4 because the step function has shifted 4 to the right."},{"Start":"00:36.120 ","End":"00:44.280","Text":"Now, this is a special case of f(t minus 4) u(t minus 4),"},{"Start":"00:44.280 ","End":"00:45.680","Text":"but I\u0027m getting ahead of myself."},{"Start":"00:45.680 ","End":"00:49.915","Text":"Let\u0027s take it slower. We want the Laplace transform of g(t)."},{"Start":"00:49.915 ","End":"00:53.705","Text":"g(t) is just this so we put that instead."},{"Start":"00:53.705 ","End":"00:57.140","Text":"Now there\u0027s a formula in the table of"},{"Start":"00:57.140 ","End":"01:04.085","Text":"Laplace transforms that tells us the Laplace transform of f(t minus k) u(t minus k)."},{"Start":"01:04.085 ","End":"01:06.530","Text":"The question is, can we bring it to this form?"},{"Start":"01:06.530 ","End":"01:15.755","Text":"Well, looks like if we let k=4 and then we let f be the function squared,"},{"Start":"01:15.755 ","End":"01:19.190","Text":"say f(t) equals t^2,"},{"Start":"01:19.190 ","End":"01:28.248","Text":"then say that what we have here is f(t minus 4) times u(t minus 4)."},{"Start":"01:28.248 ","End":"01:31.060","Text":"Because if f(t) is t^2,"},{"Start":"01:31.060 ","End":"01:36.590","Text":"then f(t minus 4) is t minus 4 squared. I\u0027ll go over that again."},{"Start":"01:36.590 ","End":"01:38.810","Text":"Well, I can replace k by 4."},{"Start":"01:38.810 ","End":"01:40.490","Text":"That\u0027s the easy part."},{"Start":"01:40.490 ","End":"01:45.705","Text":"But now, I say that if f(t) is t^2,"},{"Start":"01:45.705 ","End":"01:51.350","Text":"then f(t minus 4) is going to be t minus 4 squared,"},{"Start":"01:51.350 ","End":"01:53.510","Text":"which is what I have here."},{"Start":"01:53.510 ","End":"01:56.915","Text":"We do have f(t minus 4) u(t minus 4),"},{"Start":"01:56.915 ","End":"01:59.714","Text":"then we can apply this formula."},{"Start":"01:59.714 ","End":"02:03.650","Text":"Here, I replace f(t) by t^2."},{"Start":"02:03.650 ","End":"02:05.695","Text":"f(t) is t^2."},{"Start":"02:05.695 ","End":"02:07.640","Text":"The question is,"},{"Start":"02:07.640 ","End":"02:11.880","Text":"what is the Laplace transform of t^2, and that,"},{"Start":"02:11.880 ","End":"02:17.330","Text":"we can get by looking it up in the table of Laplace transforms."},{"Start":"02:17.330 ","End":"02:20.000","Text":"You might not have exactly t^2 in the table,"},{"Start":"02:20.000 ","End":"02:22.625","Text":"but you\u0027ll have t^n. We need n!"},{"Start":"02:22.625 ","End":"02:24.620","Text":"over s to the n minus 1,"},{"Start":"02:24.620 ","End":"02:26.600","Text":"and if we put n=2,"},{"Start":"02:26.600 ","End":"02:28.370","Text":"then we will get 2 factorial,"},{"Start":"02:28.370 ","End":"02:31.535","Text":"which is 2 over,"},{"Start":"02:31.535 ","End":"02:34.460","Text":"s to the 2 plus 1 is s^3."},{"Start":"02:34.460 ","End":"02:36.440","Text":"That\u0027s this part, the L part."},{"Start":"02:36.440 ","End":"02:39.955","Text":"We still have e^minus 4s times this."},{"Start":"02:39.955 ","End":"02:47.610","Text":"Then I\u0027ll just slightly rewrite it as this and this is our answer. We\u0027re done."}],"ID":7916},{"Watched":false,"Name":"Exercise 23","Duration":"7m 39s","ChapterTopicVideoID":7865,"CourseChapterTopicPlaylistID":4246,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"In this exercise, we\u0027re going to compute"},{"Start":"00:02.190 ","End":"00:05.915","Text":"the Laplace transform of the following piece-wise function."},{"Start":"00:05.915 ","End":"00:09.920","Text":"We\u0027re going do it in three different ways."},{"Start":"00:09.920 ","End":"00:13.110","Text":"Let\u0027s start with the first way of doing it."},{"Start":"00:13.110 ","End":"00:16.350","Text":"What I can do, similar to the previous exercise,"},{"Start":"00:16.350 ","End":"00:18.885","Text":"is to take something out the brackets."},{"Start":"00:18.885 ","End":"00:24.495","Text":"I can take t^2 up in front and that leaves here 1 and here is still 0."},{"Start":"00:24.495 ","End":"00:26.940","Text":"I mean, me check t^2 times 0 is 0,"},{"Start":"00:26.940 ","End":"00:28.935","Text":"t^2 times 1 is t^2."},{"Start":"00:28.935 ","End":"00:31.949","Text":"Now this is the shifted step function,"},{"Start":"00:31.949 ","End":"00:34.470","Text":"even seen it before in the previous exercise."},{"Start":"00:34.470 ","End":"00:37.245","Text":"It\u0027s step function shifted 4 to the right,"},{"Start":"00:37.245 ","End":"00:41.205","Text":"which we can write as u(t) minus 4."},{"Start":"00:41.205 ","End":"00:45.440","Text":"Now I can recall one of the rules we had for Laplace transforms,"},{"Start":"00:45.440 ","End":"00:49.055","Text":"is that if we have t to some power times the function,"},{"Start":"00:49.055 ","End":"00:53.900","Text":"we can compute the Laplace transform of that using this formula where"},{"Start":"00:53.900 ","End":"00:59.030","Text":"this n is nth derivative or how we can apply it in our case."},{"Start":"00:59.030 ","End":"01:02.130","Text":"Now we have t^2, u(t) minus 4."},{"Start":"01:02.130 ","End":"01:04.115","Text":"I think this looks like this."},{"Start":"01:04.115 ","End":"01:09.395","Text":"If we let n=2 and if we let"},{"Start":"01:09.395 ","End":"01:16.295","Text":"f(t) be the shifted step function, u(t) minus 4."},{"Start":"01:16.295 ","End":"01:18.289","Text":"Now following this recipe,"},{"Start":"01:18.289 ","End":"01:20.253","Text":"what we have is minus 1^n,"},{"Start":"01:20.253 ","End":"01:25.150","Text":"which is 2 and then we have the Laplace transform of f(t)."},{"Start":"01:25.150 ","End":"01:30.490","Text":"But f(t), we said was u(t) minus 4 and if n is 2 here,"},{"Start":"01:30.490 ","End":"01:33.200","Text":"second derivative, I\u0027ll write it as prime,"},{"Start":"01:33.200 ","End":"01:36.005","Text":"prime means derived twice."},{"Start":"01:36.005 ","End":"01:41.839","Text":"Now we\u0027ve already learned about the Laplace transform of a shifted unit step function."},{"Start":"01:41.839 ","End":"01:43.850","Text":"This gives me this."},{"Start":"01:43.850 ","End":"01:49.370","Text":"There is a general rule for u(t) minus k, which gives us,"},{"Start":"01:49.370 ","End":"01:53.780","Text":"if we do the Laplace transform e^-ks/s,"},{"Start":"01:53.780 ","End":"01:55.115","Text":"maybe that rings a bell."},{"Start":"01:55.115 ","End":"01:56.900","Text":"Any event with k=4 here,"},{"Start":"01:56.900 ","End":"01:58.070","Text":"this is what we get,"},{"Start":"01:58.070 ","End":"02:02.050","Text":"but we still have to do the double derivative."},{"Start":"02:02.050 ","End":"02:05.975","Text":"Then now you know how to differentiate and even to differentiate twice."},{"Start":"02:05.975 ","End":"02:11.705","Text":"It turns out to be this after simplification and this is the answer."},{"Start":"02:11.705 ","End":"02:14.750","Text":"But remember we going to do it in three different ways."},{"Start":"02:14.750 ","End":"02:16.475","Text":"This is way number one."},{"Start":"02:16.475 ","End":"02:20.390","Text":"You know what, I\u0027ll highlight it like so and then later you can refer to"},{"Start":"02:20.390 ","End":"02:24.530","Text":"it and see that we get the same answer in the two other ways we do this problem."},{"Start":"02:24.530 ","End":"02:27.680","Text":"Here we are with the second method of solving it."},{"Start":"02:27.680 ","End":"02:33.300","Text":"Now the first steps I took from the first method that we took the t^2"},{"Start":"02:33.300 ","End":"02:35.990","Text":"out the brackets and then wrote this in terms of"},{"Start":"02:35.990 ","End":"02:39.145","Text":"the unit step function shifted 4 units to the right."},{"Start":"02:39.145 ","End":"02:41.420","Text":"At this point I\u0027m going to be different."},{"Start":"02:41.420 ","End":"02:43.700","Text":"There we use one formula."},{"Start":"02:43.700 ","End":"02:48.175","Text":"Here, we\u0027re going to use another formula and the rule for Laplace transforms."},{"Start":"02:48.175 ","End":"02:51.340","Text":"This is the rule we had this if you recall for"},{"Start":"02:51.340 ","End":"02:56.210","Text":"the shifted and truncated function k units to the right,"},{"Start":"02:56.210 ","End":"02:57.845","Text":"we have this formula."},{"Start":"02:57.845 ","End":"03:02.330","Text":"The thing is, I\u0027m not quite sure what to let f be."},{"Start":"03:02.330 ","End":"03:08.210","Text":"Now, clearly we\u0027re going to let k=4 and then this bit is that bit."},{"Start":"03:08.210 ","End":"03:12.350","Text":"But what is f in order to get t^2?"},{"Start":"03:12.350 ","End":"03:15.065","Text":"Because here I have t minus 4."},{"Start":"03:15.065 ","End":"03:19.190","Text":"I mean, I have to find a function that when I substitute instead of t,"},{"Start":"03:19.190 ","End":"03:22.040","Text":"t minus 4, I get t^2."},{"Start":"03:22.040 ","End":"03:23.690","Text":"It\u0027s a backward problem,"},{"Start":"03:23.690 ","End":"03:24.995","Text":"but here\u0027s the trick."},{"Start":"03:24.995 ","End":"03:28.445","Text":"What I can do is write this g instead of t^2,"},{"Start":"03:28.445 ","End":"03:31.520","Text":"I can write it as t minus 4+4."},{"Start":"03:31.520 ","End":"03:34.490","Text":"Write it as t minus 7+7."},{"Start":"03:34.490 ","End":"03:39.580","Text":"The reason I\u0027m doing it this way is I want this t minus 4 look at it as a unit."},{"Start":"03:39.580 ","End":"03:42.110","Text":"I have t minus 4 here and I want to find"},{"Start":"03:42.110 ","End":"03:46.910","Text":"the function that just takes t instead of t minus 4."},{"Start":"03:46.910 ","End":"03:49.115","Text":"In short, I\u0027ll just write it for you."},{"Start":"03:49.115 ","End":"03:53.855","Text":"Look, if I define f(t) to be t+4^2,"},{"Start":"03:53.855 ","End":"04:00.765","Text":"then f(t) minus 4 is what I get when I replace t by t minus 4,"},{"Start":"04:00.765 ","End":"04:05.445","Text":"and so the t minus 4+4 ends up being just t^2,"},{"Start":"04:05.445 ","End":"04:07.250","Text":"so that\u0027s the idea."},{"Start":"04:07.250 ","End":"04:11.480","Text":"Now, we do have it in the form we want and so now we can"},{"Start":"04:11.480 ","End":"04:17.210","Text":"apply the rule that\u0027s over here to say that L of our function,"},{"Start":"04:17.210 ","End":"04:24.275","Text":"which is g(t) with u(t) minus 4 f(t) minus 4 is by this rule this,"},{"Start":"04:24.275 ","End":"04:28.755","Text":"and remember f(t) is t+4^2 there."},{"Start":"04:28.755 ","End":"04:34.830","Text":"Now I want to just expand the t+4^2 using binomial expansion."},{"Start":"04:34.830 ","End":"04:40.535","Text":"This is what we get t^2 plus twice t times 4 is 8t+4^2 is 16."},{"Start":"04:40.535 ","End":"04:44.960","Text":"Now I\u0027m going to break this transform up by linearity of the transform."},{"Start":"04:44.960 ","End":"04:47.345","Text":"But I need to know the building blocks."},{"Start":"04:47.345 ","End":"04:49.210","Text":"Here, it\u0027s something times t^2,"},{"Start":"04:49.210 ","End":"04:50.640","Text":"well, 1 times t^2."},{"Start":"04:50.640 ","End":"04:53.240","Text":"Here we have constant times t,"},{"Start":"04:53.240 ","End":"04:55.265","Text":"and here I have a constant times 1."},{"Start":"04:55.265 ","End":"04:58.640","Text":"I need the Laplace transform for each of these three."},{"Start":"04:58.640 ","End":"05:01.415","Text":"That\u0027s where this formula comes in."},{"Start":"05:01.415 ","End":"05:05.330","Text":"L(t)^n is n factorial s^n+1 and we could write"},{"Start":"05:05.330 ","End":"05:09.800","Text":"separately that L of the function 1 is 1/S."},{"Start":"05:09.800 ","End":"05:12.305","Text":"Then as I said, using the linearity,"},{"Start":"05:12.305 ","End":"05:15.690","Text":"what we get is as follows, e^-4s I just copy."},{"Start":"05:15.690 ","End":"05:20.115","Text":"Now l(t)^2 from this formula is 2/s^2."},{"Start":"05:20.115 ","End":"05:25.710","Text":"Here we get 8t and t is 1/s^2 and transform and the 8 is there,"},{"Start":"05:25.710 ","End":"05:28.520","Text":"and here\u0027s the 16 with a 1/s."},{"Start":"05:28.520 ","End":"05:31.790","Text":"Now this doesn\u0027t look quite the same as what we had before,"},{"Start":"05:31.790 ","End":"05:36.470","Text":"but it\u0027s simplifies to this and this is exactly what we had before."},{"Start":"05:36.470 ","End":"05:39.845","Text":"Just go back and check and see that this is what we had."},{"Start":"05:39.845 ","End":"05:42.545","Text":"That was way number 2."},{"Start":"05:42.545 ","End":"05:44.480","Text":"Now we\u0027re going to do it the third way,"},{"Start":"05:44.480 ","End":"05:47.285","Text":"which will be directly from the definition."},{"Start":"05:47.285 ","End":"05:49.085","Text":"Directly off the definition,"},{"Start":"05:49.085 ","End":"05:52.025","Text":"perhaps I should have copied g(t) again."},{"Start":"05:52.025 ","End":"05:56.990","Text":"G(t), remember was equal to either 0"},{"Start":"05:56.990 ","End":"06:04.195","Text":"or t^2 according to whether t was less than 4 or greater or equal to 4."},{"Start":"06:04.195 ","End":"06:07.550","Text":"Now the integral, I could take it from 0 to infinity,"},{"Start":"06:07.550 ","End":"06:10.320","Text":"but the part from 0-4 is just 0,"},{"Start":"06:10.320 ","End":"06:13.110","Text":"and that\u0027s why I only take it from 4 to infinity."},{"Start":"06:13.110 ","End":"06:21.315","Text":"Then this t^2 is this t^2 because it\u0027s 0."},{"Start":"06:21.315 ","End":"06:25.505","Text":"When it\u0027s less than 4 that\u0027s why we take the integral from 4."},{"Start":"06:25.505 ","End":"06:28.730","Text":"I\u0027m going to spare you the details of the integral."},{"Start":"06:28.730 ","End":"06:31.565","Text":"We do it by parts twice."},{"Start":"06:31.565 ","End":"06:37.670","Text":"I\u0027ll leave you that to check that this is the integral of this function with respect to"},{"Start":"06:37.670 ","End":"06:44.390","Text":"t. Now we just have to substitute t=4 and t equals infinity."},{"Start":"06:44.390 ","End":"06:47.270","Text":"It\u0027s an improper integral."},{"Start":"06:47.270 ","End":"06:50.060","Text":"We put t as something large which tends to infinity,"},{"Start":"06:50.060 ","End":"06:52.385","Text":"but it\u0027s as if we put infinity in."},{"Start":"06:52.385 ","End":"06:55.850","Text":"As I often do we make the assumption that s"},{"Start":"06:55.850 ","End":"06:59.930","Text":"is in some range and that way the improper integral converges."},{"Start":"06:59.930 ","End":"07:03.260","Text":"If s is positive when t goes to infinity,"},{"Start":"07:03.260 ","End":"07:07.275","Text":"e^-st, e to the minus infinity."},{"Start":"07:07.275 ","End":"07:10.495","Text":"This part is 0."},{"Start":"07:10.495 ","End":"07:15.810","Text":"Then that\u0027s the 0. We subtract the t=4 part, that\u0027s easy."},{"Start":"07:15.810 ","End":"07:18.885","Text":"Instead of t^2 I put 16, instead of 2_t,"},{"Start":"07:18.885 ","End":"07:21.350","Text":"I put 8, and this is what we get,"},{"Start":"07:21.350 ","End":"07:24.305","Text":"then the minus times a minus is a plus."},{"Start":"07:24.305 ","End":"07:26.225","Text":"If we simplify this,"},{"Start":"07:26.225 ","End":"07:31.070","Text":"divide everything by x^3 then we can get it into this shape."},{"Start":"07:31.070 ","End":"07:33.755","Text":"That is exactly what we had before."},{"Start":"07:33.755 ","End":"07:36.650","Text":"That\u0027s the third method of solving it and all three methods"},{"Start":"07:36.650 ","End":"07:40.530","Text":"led to the same answer and we are done."}],"ID":7917}],"Thumbnail":null,"ID":4246},{"Name":"The Inverse Laplace Transform","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"14s","ChapterTopicVideoID":7931,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.375","Text":"Here we have to find the inverse Laplace transform of 1/s."},{"Start":"00:06.375 ","End":"00:09.000","Text":"We just look it up in the table of inverse"},{"Start":"00:09.000 ","End":"00:15.490","Text":"transforms where it appears and the answer is 1. That\u0027s it."}],"ID":8003},{"Watched":false,"Name":"Exercise 2","Duration":"25s","ChapterTopicVideoID":7932,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"Here, we have to find the inverse Laplace transform of 1 over s^4."},{"Start":"00:04.755 ","End":"00:07.770","Text":"We go to the table of inverse transforms,"},{"Start":"00:07.770 ","End":"00:10.455","Text":"1 over s^4 is probably not in there,"},{"Start":"00:10.455 ","End":"00:14.315","Text":"but we do find 1 over s^n in general."},{"Start":"00:14.315 ","End":"00:17.970","Text":"All we have to do is substitute n=4 here."},{"Start":"00:17.970 ","End":"00:20.910","Text":"This is what we get, 4 minus 1 is 3."},{"Start":"00:20.910 ","End":"00:25.990","Text":"Of course, you could write it t^3 over 6. Anyway, that\u0027s it."}],"ID":8004},{"Watched":false,"Name":"Exercise 3","Duration":"20s","ChapterTopicVideoID":7933,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.265","Text":"Here, we have to compute the Inverse Laplace Transform of 1 over s minus 10."},{"Start":"00:05.265 ","End":"00:08.550","Text":"We go to the table for inverse transforms and the"},{"Start":"00:08.550 ","End":"00:12.405","Text":"closest we find is this 1 over s minus a."},{"Start":"00:12.405 ","End":"00:16.680","Text":"All we have to do is substitute a=10 here and this"},{"Start":"00:16.680 ","End":"00:21.790","Text":"gives us the answer of e^10t. We\u0027re done."}],"ID":8005},{"Watched":false,"Name":"Exercise 4","Duration":"23s","ChapterTopicVideoID":7934,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"Here, we have to find the Inverse Laplace Transform of 1 over s^2 plus 4."},{"Start":"00:04.575 ","End":"00:11.385","Text":"We go to the table of inverse transforms and the closest we can find to this,"},{"Start":"00:11.385 ","End":"00:16.545","Text":"is this entry and that\u0027s just fine for us because if we let a equals 2,"},{"Start":"00:16.545 ","End":"00:24.040","Text":"then 2^2 is 4 and we will get 1/2 sine 2t, that\u0027s it."}],"ID":8006},{"Watched":false,"Name":"Exercise 5","Duration":"21s","ChapterTopicVideoID":7935,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"Here we want to compute the Inverse Laplace Transform of s/s^2 plus 4."},{"Start":"00:06.285 ","End":"00:10.135","Text":"We go to our lookup table of inverse transforms,"},{"Start":"00:10.135 ","End":"00:12.540","Text":"and the closest match is this,"},{"Start":"00:12.540 ","End":"00:14.385","Text":"which looks very much like this."},{"Start":"00:14.385 ","End":"00:17.145","Text":"If we just let a=2,"},{"Start":"00:17.145 ","End":"00:22.530","Text":"so that gives us that the answer is just cosine 2t. That\u0027s it."}],"ID":8007},{"Watched":false,"Name":"Exercise 6","Duration":"1m 50s","ChapterTopicVideoID":7936,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"Here, we have to compute the inverse plus transform of this function here."},{"Start":"00:05.550 ","End":"00:10.245","Text":"I\u0027ll give it a name, I\u0027ll call this big F(s)."},{"Start":"00:10.245 ","End":"00:13.170","Text":"Now, in my table of inverse transforms,"},{"Start":"00:13.170 ","End":"00:17.295","Text":"the closest I can find to this is this,"},{"Start":"00:17.295 ","End":"00:20.085","Text":"and if I let a equals 2,"},{"Start":"00:20.085 ","End":"00:22.320","Text":"I don\u0027t get exactly this."},{"Start":"00:22.320 ","End":"00:24.720","Text":"I get the inverse transform of this,"},{"Start":"00:24.720 ","End":"00:25.770","Text":"and the question is,"},{"Start":"00:25.770 ","End":"00:28.395","Text":"how do I go from s,"},{"Start":"00:28.395 ","End":"00:32.040","Text":"which is here, to s minus 10, which is here."},{"Start":"00:32.040 ","End":"00:34.005","Text":"How do I bridge that gap?"},{"Start":"00:34.005 ","End":"00:35.955","Text":"Well, we use rules for this."},{"Start":"00:35.955 ","End":"00:43.985","Text":"One of the rules is that if you have a function of s and you want to add a number to s,"},{"Start":"00:43.985 ","End":"00:45.665","Text":"then you can do that."},{"Start":"00:45.665 ","End":"00:46.980","Text":"Let\u0027s say you want to add a,"},{"Start":"00:46.980 ","End":"00:50.120","Text":"as long as you put e^at in front."},{"Start":"00:50.120 ","End":"00:52.655","Text":"Or in our case, if we let a equals 10,"},{"Start":"00:52.655 ","End":"00:58.625","Text":"it says that if I replace s by s plus 10, that\u0027s okay."},{"Start":"00:58.625 ","End":"01:03.725","Text":"But I have to compensate by sticking an e to the 10^t in front."},{"Start":"01:03.725 ","End":"01:08.885","Text":"Let\u0027s see what happens if we do compute F(s) plus 10 here."},{"Start":"01:08.885 ","End":"01:12.320","Text":"I take this expression where I see s,"},{"Start":"01:12.320 ","End":"01:16.100","Text":"I replace it with s plus 10."},{"Start":"01:16.100 ","End":"01:18.365","Text":"Where I see s here,"},{"Start":"01:18.365 ","End":"01:21.260","Text":"I replace it by s plus 10,"},{"Start":"01:21.260 ","End":"01:27.470","Text":"and then I compensate by putting e to the 10^t here, just like here."},{"Start":"01:27.470 ","End":"01:32.840","Text":"Now, this is good because s plus 10 minus 10 is just s."},{"Start":"01:32.840 ","End":"01:39.420","Text":"That gives us e to the 10^t the inverse transform of 1 over s^2 plus 4."},{"Start":"01:39.420 ","End":"01:41.180","Text":"But we had that in the beginning,"},{"Start":"01:41.180 ","End":"01:45.165","Text":"we saw that this was just 1/2 sine 2t."},{"Start":"01:45.165 ","End":"01:48.435","Text":"Then we have the extra e to the 10^t in front of it,"},{"Start":"01:48.435 ","End":"01:51.220","Text":"and that\u0027s the answer. We\u0027re done."}],"ID":8008},{"Watched":false,"Name":"Exercise 7","Duration":"3m 15s","ChapterTopicVideoID":7937,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"Here, we have to find the Inverse Laplace Transform"},{"Start":"00:02.865 ","End":"00:05.937","Text":"of this function and I\u0027ll give it a name,"},{"Start":"00:05.937 ","End":"00:08.925","Text":"I\u0027ll call it f(s)."},{"Start":"00:08.925 ","End":"00:12.330","Text":"We go to the table of inverse transforms."},{"Start":"00:12.330 ","End":"00:14.930","Text":"The closest I can find is this,"},{"Start":"00:14.930 ","End":"00:18.330","Text":"but there are more advanced tables and you might find a closer match."},{"Start":"00:18.330 ","End":"00:20.280","Text":"Anyway, this is what I can find."},{"Start":"00:20.280 ","End":"00:23.310","Text":"If I let a=2,"},{"Start":"00:23.310 ","End":"00:26.580","Text":"then I have that the inverse transform of s over s^2 plus"},{"Start":"00:26.580 ","End":"00:30.555","Text":"4 is cosine 2 t. But that\u0027s not what I have."},{"Start":"00:30.555 ","End":"00:38.580","Text":"I need a way from getting from s and somehow replacing it with s minus 2."},{"Start":"00:38.580 ","End":"00:41.340","Text":"Now fortunately, besides entries,"},{"Start":"00:41.340 ","End":"00:43.118","Text":"there are also rules,"},{"Start":"00:43.118 ","End":"00:45.720","Text":"and the rule that\u0027s going to help us is this one which"},{"Start":"00:45.720 ","End":"00:48.750","Text":"says that if I have a function of s,"},{"Start":"00:48.750 ","End":"00:51.090","Text":"then I\u0027m looking for its inverse transform."},{"Start":"00:51.090 ","End":"00:54.947","Text":"But if instead of s, I put s plus a, that\u0027s okay,"},{"Start":"00:54.947 ","End":"00:59.220","Text":"but I have to compensate by putting e^at in front."},{"Start":"00:59.220 ","End":"01:02.925","Text":"In our case, if I let a=2,"},{"Start":"01:02.925 ","End":"01:07.350","Text":"this 2 is not the same as the 2 from here,"},{"Start":"01:07.350 ","End":"01:09.060","Text":"it\u0027s a different formula,"},{"Start":"01:09.060 ","End":"01:12.090","Text":"that\u0027s the 2 from here that I\u0027m taking."},{"Start":"01:12.090 ","End":"01:15.915","Text":"Just by chance, there\u0027s 2 and then here 2^2."},{"Start":"01:15.915 ","End":"01:19.485","Text":"Different a, but just happens to be the same value."},{"Start":"01:19.485 ","End":"01:23.190","Text":"In our case, the rule becomes this."},{"Start":"01:23.190 ","End":"01:28.725","Text":"I\u0027m going to use our function s and replace s with s plus 2."},{"Start":"01:28.725 ","End":"01:32.835","Text":"What I get is instead of this s here,"},{"Start":"01:32.835 ","End":"01:34.590","Text":"I have s plus 2,"},{"Start":"01:34.590 ","End":"01:36.390","Text":"instead of this s,"},{"Start":"01:36.390 ","End":"01:38.535","Text":"I also have s plus 2,"},{"Start":"01:38.535 ","End":"01:41.355","Text":"and that\u0027s f(s) plus 2,"},{"Start":"01:41.355 ","End":"01:46.729","Text":"but I also have to compensate by putting this e^2t in front."},{"Start":"01:46.729 ","End":"01:49.300","Text":"Now we can simplify this."},{"Start":"01:49.300 ","End":"01:55.005","Text":"s plus 2 minus 2 is just s. The denominator simplifies a lot."},{"Start":"01:55.005 ","End":"01:58.465","Text":"We have s plus 2 over s^2 plus 4."},{"Start":"01:58.465 ","End":"02:01.880","Text":"Now how do we find the inverse transform of this?"},{"Start":"02:01.880 ","End":"02:04.580","Text":"I break it up into 2, the s plus 2."},{"Start":"02:04.580 ","End":"02:07.670","Text":"I put the s separately and the 2 separately."},{"Start":"02:07.670 ","End":"02:12.035","Text":"We\u0027re going to use the linearity of the inverse transform."},{"Start":"02:12.035 ","End":"02:15.230","Text":"This is the rule for linearity in general,"},{"Start":"02:15.230 ","End":"02:16.730","Text":"that\u0027s a bit abstract."},{"Start":"02:16.730 ","End":"02:18.890","Text":"In our case, it just means that I can apply"},{"Start":"02:18.890 ","End":"02:22.205","Text":"the inverse transform to this bit that\u0027s here,"},{"Start":"02:22.205 ","End":"02:23.623","Text":"and also to this bit,"},{"Start":"02:23.623 ","End":"02:27.395","Text":"and also take the 2 outside the transform."},{"Start":"02:27.395 ","End":"02:30.920","Text":"This thing just rides along."},{"Start":"02:30.920 ","End":"02:36.110","Text":"We\u0027re in good shape because we can find these both in the table."},{"Start":"02:36.110 ","End":"02:39.785","Text":"Well, actually this bit we already did earlier, if you look back,"},{"Start":"02:39.785 ","End":"02:43.205","Text":"and this comes out to be cosine 2t,"},{"Start":"02:43.205 ","End":"02:45.245","Text":"just need to look up this."},{"Start":"02:45.245 ","End":"02:46.910","Text":"I find this rule,"},{"Start":"02:46.910 ","End":"02:50.770","Text":"which is just what we have if we let a=2 here."},{"Start":"02:50.770 ","End":"02:54.560","Text":"This inverse transform is 1/2 sine 2t."},{"Start":"02:54.560 ","End":"02:57.871","Text":"Now all I have to do is take this plus twice this,"},{"Start":"02:57.871 ","End":"03:01.340","Text":"and that gives us this, the e^2t stays."},{"Start":"03:01.340 ","End":"03:05.765","Text":"This part\u0027s the cosine 2t and 2 with the 1/2 cancel,"},{"Start":"03:05.765 ","End":"03:11.660","Text":"I guess we could just cancel that one and I just erase it and it\u0027ll be much neater."},{"Start":"03:11.660 ","End":"03:15.480","Text":"There we go. That\u0027s the answer."}],"ID":8009},{"Watched":false,"Name":"Exercise 8","Duration":"24s","ChapterTopicVideoID":7938,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here, we want to compute the Inverse Laplace Transform of this expression."},{"Start":"00:04.830 ","End":"00:08.010","Text":"As usual, we go to the table of"},{"Start":"00:08.010 ","End":"00:11.760","Text":"inverse transforms and we find something very close to this."},{"Start":"00:11.760 ","End":"00:15.319","Text":"All I have to do really is put a equals 2, of course,"},{"Start":"00:15.319 ","End":"00:18.390","Text":"a squared is 4, and if I substitute that here,"},{"Start":"00:18.390 ","End":"00:20.370","Text":"this gives us the answer."},{"Start":"00:20.370 ","End":"00:22.703","Text":"2 times 2 is 4 and here we put the 2."},{"Start":"00:22.703 ","End":"00:24.910","Text":"That\u0027s it."}],"ID":8010},{"Watched":false,"Name":"Exercise 9","Duration":"22s","ChapterTopicVideoID":7939,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Here, we want to compute the Inverse Laplace Transform of"},{"Start":"00:03.570 ","End":"00:07.304","Text":"this function of s. We go to the table,"},{"Start":"00:07.304 ","End":"00:09.465","Text":"we find something very close."},{"Start":"00:09.465 ","End":"00:11.460","Text":"Only differences here, a^2 here 4,"},{"Start":"00:11.460 ","End":"00:15.000","Text":"so a equals 2 because 2^2 is 4, and substitute,"},{"Start":"00:15.000 ","End":"00:16.303","Text":"and this is what we get,"},{"Start":"00:16.303 ","End":"00:22.750","Text":"except that you might want to simplify it and replace 2 times 2 cubed by 16. We\u0027re done."}],"ID":8011},{"Watched":false,"Name":"Exercise 10","Duration":"41s","ChapterTopicVideoID":7940,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"Here, we want to compute the Inverse Laplace Transform of 1 over the square root of"},{"Start":"00:04.770 ","End":"00:10.530","Text":"s. We go to the table of inverse transforms and we don\u0027t have this exactly,"},{"Start":"00:10.530 ","End":"00:12.089","Text":"we have something close."},{"Start":"00:12.089 ","End":"00:14.445","Text":"We have the inverse transform of this."},{"Start":"00:14.445 ","End":"00:17.640","Text":"Now, the only difference between this and this is we have this extra constant"},{"Start":"00:17.640 ","End":"00:21.210","Text":"thrown in and we can deal with constants using the linearity."},{"Start":"00:21.210 ","End":"00:24.720","Text":"What we do is we adjust this or rather"},{"Start":"00:24.720 ","End":"00:28.380","Text":"we take the constant outside of the inverse transform,"},{"Start":"00:28.380 ","End":"00:31.410","Text":"which is linear, and so all that"},{"Start":"00:31.410 ","End":"00:35.445","Text":"remains to do now is to divide both sides by this constant,"},{"Start":"00:35.445 ","End":"00:38.265","Text":"which then goes to the denominator here."},{"Start":"00:38.265 ","End":"00:41.590","Text":"This is our answer, and we\u0027re done."}],"ID":8012},{"Watched":false,"Name":"Exercise 11","Duration":"2m 44s","ChapterTopicVideoID":7941,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.045","Text":"In this exercise, we want to compute the Inverse Laplace Transform of 1 over s^2 minus 4."},{"Start":"00:06.045 ","End":"00:08.340","Text":"Now, in my table of inverse transforms,"},{"Start":"00:08.340 ","End":"00:10.770","Text":"I don\u0027t find anything similar to this."},{"Start":"00:10.770 ","End":"00:15.465","Text":"There are extended tables and you might find something there but not in mine."},{"Start":"00:15.465 ","End":"00:19.410","Text":"So what I\u0027m going to do is use a technique of"},{"Start":"00:19.410 ","End":"00:25.305","Text":"partial fractions to decompose this into pieces which are more manageable."},{"Start":"00:25.305 ","End":"00:27.375","Text":"Notice that the denominator,"},{"Start":"00:27.375 ","End":"00:30.269","Text":"using the difference of squares formula from algebra,"},{"Start":"00:30.269 ","End":"00:33.585","Text":"I can write as s minus 2 times s plus 2."},{"Start":"00:33.585 ","End":"00:37.995","Text":"Now, we want to treat this as a partial fractions exercise."},{"Start":"00:37.995 ","End":"00:41.730","Text":"I mean, this part I want to decompose into partial fractions."},{"Start":"00:41.730 ","End":"00:43.920","Text":"We know that the general form is going to be something"},{"Start":"00:43.920 ","End":"00:46.380","Text":"over s minus 2 and some other thing,"},{"Start":"00:46.380 ","End":"00:53.970","Text":"I mean constants over s plus 2 and multiply both sides by s minus 2s plus 2,"},{"Start":"00:53.970 ","End":"00:57.075","Text":"the common denominator, and then we get rid of the fractions,"},{"Start":"00:57.075 ","End":"01:00.200","Text":"and so we\u0027ve got this equation."},{"Start":"01:00.200 ","End":"01:02.509","Text":"It\u0027s not an equation,"},{"Start":"01:02.509 ","End":"01:07.555","Text":"it\u0027s got to be true for all s. We can substitute whatever values of s we like,"},{"Start":"01:07.555 ","End":"01:13.370","Text":"and the usual technique we tend to substitute something that makes a factor 0,"},{"Start":"01:13.370 ","End":"01:15.065","Text":"like if I let s equals 2,"},{"Start":"01:15.065 ","End":"01:22.285","Text":"this thing becomes 0 and I\u0027ve got 1 equals A times 2 plus 2 plus B times 0 here."},{"Start":"01:22.285 ","End":"01:25.415","Text":"4A is 1, A is 1/4."},{"Start":"01:25.415 ","End":"01:28.565","Text":"Similarly, if I let s equal minus 2,"},{"Start":"01:28.565 ","End":"01:30.350","Text":"this thing becomes 0,"},{"Start":"01:30.350 ","End":"01:32.285","Text":"this becomes minus 4,"},{"Start":"01:32.285 ","End":"01:36.510","Text":"we get an equation that 1 equals minus 4B,"},{"Start":"01:36.510 ","End":"01:38.280","Text":"so B is minus a quarter,"},{"Start":"01:38.280 ","End":"01:40.920","Text":"and then I can put these 2 in here,"},{"Start":"01:40.920 ","End":"01:43.730","Text":"and getting back to inverse transform,"},{"Start":"01:43.730 ","End":"01:46.670","Text":"all this was inside the inverse transform Laplace,"},{"Start":"01:46.670 ","End":"01:47.885","Text":"so we\u0027ve got this."},{"Start":"01:47.885 ","End":"01:50.720","Text":"Now we\u0027re going to use linearity of the inverse transform to"},{"Start":"01:50.720 ","End":"01:53.990","Text":"break it up into 2 pieces and take constants out."},{"Start":"01:53.990 ","End":"01:58.085","Text":"The precise rule for linearity is like this,"},{"Start":"01:58.085 ","End":"02:00.200","Text":"but you know how to work it."},{"Start":"02:00.200 ","End":"02:01.910","Text":"In our case, for example,"},{"Start":"02:01.910 ","End":"02:03.800","Text":"we split it up into 2,"},{"Start":"02:03.800 ","End":"02:07.880","Text":"but this plus becomes a minus because we take the constants out,"},{"Start":"02:07.880 ","End":"02:09.425","Text":"the quarter goes in front,"},{"Start":"02:09.425 ","End":"02:12.185","Text":"the minus quarter goes in front here."},{"Start":"02:12.185 ","End":"02:17.345","Text":"Now, this is very good because we do have in the table,"},{"Start":"02:17.345 ","End":"02:19.810","Text":"not exactly 1 over s minus 2,"},{"Start":"02:19.810 ","End":"02:25.585","Text":"but we do have the inverse transform of 1 over s minus a and also 1 over s plus a."},{"Start":"02:25.585 ","End":"02:29.367","Text":"This really is the same rule as this, but anyway."},{"Start":"02:29.367 ","End":"02:32.915","Text":"In both of these I\u0027ll put s equals 2,"},{"Start":"02:32.915 ","End":"02:37.440","Text":"and that will give us our answer 1/4 of e^at,"},{"Start":"02:37.440 ","End":"02:39.150","Text":"which is e^2t,"},{"Start":"02:39.150 ","End":"02:44.620","Text":"minus 1/4 e to the minus 2t, and we\u0027re done."}],"ID":8013},{"Watched":false,"Name":"Exercise 12","Duration":"3m 27s","ChapterTopicVideoID":7930,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this exercise, we want to compute the Inverse Laplace Transform of"},{"Start":"00:05.070 ","End":"00:10.620","Text":"this rational expression in s. 5 minus s over s^2 plus 5s."},{"Start":"00:10.620 ","End":"00:13.620","Text":"I\u0027m going to offer 2 solutions."},{"Start":"00:13.620 ","End":"00:16.605","Text":"In the first one, I\u0027m going to use partial fractions,"},{"Start":"00:16.605 ","End":"00:18.720","Text":"and then we\u0027ll see about the other."},{"Start":"00:18.720 ","End":"00:22.785","Text":"We can decompose the denominator into s times"},{"Start":"00:22.785 ","End":"00:28.770","Text":"5 plus s. This leads to a decomposition as follows."},{"Start":"00:28.770 ","End":"00:31.140","Text":"I don\u0027t know why I wrote 5 plus s here,"},{"Start":"00:31.140 ","End":"00:34.125","Text":"same thing, but it should be s plus 5."},{"Start":"00:34.125 ","End":"00:36.450","Text":"Back here to the partial fraction,"},{"Start":"00:36.450 ","End":"00:38.795","Text":"multiply by this denominator,"},{"Start":"00:38.795 ","End":"00:40.505","Text":"and this is what we get."},{"Start":"00:40.505 ","End":"00:46.385","Text":"This is true for all s. We can substitute convenient values."},{"Start":"00:46.385 ","End":"00:52.160","Text":"I\u0027m going to substitute one time s equals 0 and one time s equals minus 5."},{"Start":"00:52.160 ","End":"00:58.615","Text":"If s is 0, this disappears and we conclude that A is 1."},{"Start":"00:58.615 ","End":"01:01.010","Text":"If we let s be minus 5,"},{"Start":"01:01.010 ","End":"01:06.080","Text":"then this bit disappears and after the computation B comes out minus 2."},{"Start":"01:06.080 ","End":"01:09.525","Text":"Then we plug A and B back in here and then we get"},{"Start":"01:09.525 ","End":"01:13.760","Text":"this expression which is going to be much easier to evaluate."},{"Start":"01:13.760 ","End":"01:15.920","Text":"We can break this up using linearity."},{"Start":"01:15.920 ","End":"01:18.110","Text":"I\u0027m just quoting the formula for linearity,"},{"Start":"01:18.110 ","End":"01:20.810","Text":"but you should know how to use it by now."},{"Start":"01:20.810 ","End":"01:25.190","Text":"It basically split it up into 2 bits and pulled the minus 2 out to the second bit."},{"Start":"01:25.190 ","End":"01:31.265","Text":"We get this and then there\u0027s 2 relatively easy Laplace inverse transforms to do."},{"Start":"01:31.265 ","End":"01:33.710","Text":"We can use this formula for both of them,"},{"Start":"01:33.710 ","End":"01:37.580","Text":"once with a being 5 and once a being 0."},{"Start":"01:37.580 ","End":"01:39.260","Text":"But there\u0027s also in the table,"},{"Start":"01:39.260 ","End":"01:45.800","Text":"you might find explicitly the inverse transform of 1/s is 1, the constant function."},{"Start":"01:45.800 ","End":"01:48.755","Text":"Either way, this is the answer."},{"Start":"01:48.755 ","End":"01:51.290","Text":"Now, I\u0027m going to do it with another method,"},{"Start":"01:51.290 ","End":"01:56.405","Text":"but it assumes that you have a more extended table of Inverse Laplace Transforms."},{"Start":"01:56.405 ","End":"02:00.590","Text":"We now come to the second solution where we break this up."},{"Start":"02:00.590 ","End":"02:03.020","Text":"We don\u0027t right away factorize the denominator."},{"Start":"02:03.020 ","End":"02:07.460","Text":"We break it up as 5 over this minus s over this,"},{"Start":"02:07.460 ","End":"02:10.175","Text":"and use linearity to bring the 5 out."},{"Start":"02:10.175 ","End":"02:14.045","Text":"Now we need the Inverse Laplace Transform of these 2 bits."},{"Start":"02:14.045 ","End":"02:15.590","Text":"In this solution, I\u0027m assuming you have"},{"Start":"02:15.590 ","End":"02:21.095","Text":"a more extended table of inverse transforms and we can look both of these up."},{"Start":"02:21.095 ","End":"02:23.870","Text":"For this bit we\u0027ll be using this formula,"},{"Start":"02:23.870 ","End":"02:26.960","Text":"and for this bit we\u0027ll use this formula."},{"Start":"02:26.960 ","End":"02:29.150","Text":"In both cases, the denominator,"},{"Start":"02:29.150 ","End":"02:33.530","Text":"which is s^2 plus 5s is s(s+5)."},{"Start":"02:33.680 ","End":"02:39.885","Text":"We can see that a will be 0 and b will be minus 5,"},{"Start":"02:39.885 ","End":"02:46.425","Text":"because s is s minus 0 and s plus 5 is s minus minus 5 and here too."},{"Start":"02:46.425 ","End":"02:49.140","Text":"Just make the substitution,"},{"Start":"02:49.140 ","End":"02:54.220","Text":"a equals 0, b equals minus 5,"},{"Start":"02:54.220 ","End":"02:56.720","Text":"plug them in and we get this."},{"Start":"02:56.720 ","End":"02:59.539","Text":"Now a lot of the terms when everything cancels,"},{"Start":"02:59.539 ","End":"03:03.755","Text":"like here, 0 minus minus 5 is 5 and it cancels with this 5."},{"Start":"03:03.755 ","End":"03:06.127","Text":"Here we\u0027ve got one minus this,"},{"Start":"03:06.127 ","End":"03:07.730","Text":"here we don\u0027t have anything."},{"Start":"03:07.730 ","End":"03:11.225","Text":"Here also the 5 and the 5 cancel."},{"Start":"03:11.225 ","End":"03:16.235","Text":"In short, we get 1 minus e to the minus 5t from here,"},{"Start":"03:16.235 ","End":"03:19.790","Text":"and just minus e to the minus 5t from here."},{"Start":"03:19.790 ","End":"03:22.760","Text":"Finally, just collect these two to combine them."},{"Start":"03:22.760 ","End":"03:27.630","Text":"This is the same answer as we got before. That\u0027s it."}],"ID":8014},{"Watched":false,"Name":"Exercise 13","Duration":"4m 23s","ChapterTopicVideoID":7942,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.735","Text":"In this exercise, we have to compute the Inverse Laplace Transforms of this expression."},{"Start":"00:06.735 ","End":"00:09.915","Text":"I\u0027m going to do it in more than one way depending"},{"Start":"00:09.915 ","End":"00:13.170","Text":"on what kind of table of inverse transform you have;"},{"Start":"00:13.170 ","End":"00:16.215","Text":"a more restricted, limited one or an expanded one."},{"Start":"00:16.215 ","End":"00:20.955","Text":"But either way, we\u0027re going to begin by factorizing the denominator."},{"Start":"00:20.955 ","End":"00:24.270","Text":"It\u0027s a quadratic with leading coefficient 1."},{"Start":"00:24.270 ","End":"00:29.850","Text":"It factors into s minus s_1 say and s minus s_2,"},{"Start":"00:29.850 ","End":"00:34.590","Text":"where these are the two roots of the quadratic equation by setting it equal to 0."},{"Start":"00:34.590 ","End":"00:37.380","Text":"Then we go and solve it by whatever method you want."},{"Start":"00:37.380 ","End":"00:38.850","Text":"I\u0027m not going to do the computations here."},{"Start":"00:38.850 ","End":"00:42.390","Text":"The results are minus 3 and minus 2."},{"Start":"00:42.390 ","End":"00:47.650","Text":"Therefore, this denominator part can be written as follows."},{"Start":"00:47.650 ","End":"00:54.470","Text":"Our task now becomes to compute the Inverse Laplace Transform of this expression."},{"Start":"00:54.470 ","End":"00:57.350","Text":"As I said, it depends on what kind of table you have,"},{"Start":"00:57.350 ","End":"01:04.395","Text":"because it\u0027s possible that you\u0027ll find s/s minus a,"},{"Start":"01:04.395 ","End":"01:09.158","Text":"s minus b in the table of inverse transforms,"},{"Start":"01:09.158 ","End":"01:12.320","Text":"but if you have a more simple one, then you won\u0027t."},{"Start":"01:12.320 ","End":"01:14.390","Text":"We\u0027ll do it differently at the end,"},{"Start":"01:14.390 ","End":"01:16.280","Text":"assuming that you have an expanded table."},{"Start":"01:16.280 ","End":"01:18.140","Text":"What we need to do, in this case,"},{"Start":"01:18.140 ","End":"01:23.245","Text":"is to use partial fractions to break this up into simpler pieces."},{"Start":"01:23.245 ","End":"01:25.850","Text":"According to the theory of partial fractions,"},{"Start":"01:25.850 ","End":"01:27.980","Text":"this will decompose as follows,"},{"Start":"01:27.980 ","End":"01:30.215","Text":"where a and b are constants."},{"Start":"01:30.215 ","End":"01:34.820","Text":"What we do to find them is multiply both sides by this denominator,"},{"Start":"01:34.820 ","End":"01:37.040","Text":"which gives us this,"},{"Start":"01:37.040 ","End":"01:39.305","Text":"which is not really an equation."},{"Start":"01:39.305 ","End":"01:40.970","Text":"It\u0027s an identity really,"},{"Start":"01:40.970 ","End":"01:42.095","Text":"meaning for all s,"},{"Start":"01:42.095 ","End":"01:43.385","Text":"this has got a whole true."},{"Start":"01:43.385 ","End":"01:44.840","Text":"We have to find a and b,"},{"Start":"01:44.840 ","End":"01:46.265","Text":"these are the unknowns."},{"Start":"01:46.265 ","End":"01:49.030","Text":"We can substitute any value of s we like."},{"Start":"01:49.030 ","End":"01:50.880","Text":"For example, minus 3."},{"Start":"01:50.880 ","End":"01:52.410","Text":"Now why did I choose minus 3?"},{"Start":"01:52.410 ","End":"01:54.475","Text":"Because that will make this one 0."},{"Start":"01:54.475 ","End":"01:57.110","Text":"Then we substitute, we\u0027ll get this."},{"Start":"01:57.110 ","End":"01:59.795","Text":"We cut to the chase, A comes out to be 3."},{"Start":"01:59.795 ","End":"02:03.635","Text":"If we want this to be 0 and that s to be minus 2,"},{"Start":"02:03.635 ","End":"02:08.150","Text":"then this is what we get and we solve it and b becomes minus 2."},{"Start":"02:08.150 ","End":"02:09.860","Text":"Now that we have a and b,"},{"Start":"02:09.860 ","End":"02:12.540","Text":"we need to substitute them here."},{"Start":"02:12.540 ","End":"02:13.980","Text":"The 3 is here,"},{"Start":"02:13.980 ","End":"02:15.480","Text":"the minus 2 is here."},{"Start":"02:15.480 ","End":"02:20.230","Text":"Also I remembered the context we are in an Inverse Laplace Transform."},{"Start":"02:20.230 ","End":"02:24.960","Text":"We just substituted this by the decomposition to be this."},{"Start":"02:24.960 ","End":"02:29.755","Text":"This is much better for us because in any table of inverse transforms,"},{"Start":"02:29.755 ","End":"02:34.325","Text":"we have the inverse of 1/s plus or minus, doesn\u0027t matter."},{"Start":"02:34.325 ","End":"02:37.310","Text":"We first of all want to use the linearity property"},{"Start":"02:37.310 ","End":"02:40.745","Text":"to break it up into pieces and to pull this constants out."},{"Start":"02:40.745 ","End":"02:43.730","Text":"This is the formal definition of linearity of"},{"Start":"02:43.730 ","End":"02:47.495","Text":"the inverse transform but that\u0027s just for reference."},{"Start":"02:47.495 ","End":"02:52.375","Text":"In our case, what it means is that we can take L inverse of"},{"Start":"02:52.375 ","End":"02:58.175","Text":"1/s plus 3 and take the 3 in front and also here the minus 2 in front."},{"Start":"02:58.175 ","End":"03:04.555","Text":"Now we can use the formula for the inverse of this e to the minus at."},{"Start":"03:04.555 ","End":"03:12.545","Text":"One time we\u0027ll use this with s=3 and the other time we\u0027ll use it with s=2."},{"Start":"03:12.545 ","End":"03:15.965","Text":"We\u0027ll get this as the answer here is the 3,"},{"Start":"03:15.965 ","End":"03:17.960","Text":"here\u0027s the minus 2."},{"Start":"03:17.960 ","End":"03:22.710","Text":"The 1/s plus 3 from this formula comes e to the minus 3t here,"},{"Start":"03:22.710 ","End":"03:24.090","Text":"here to the minus 2t."},{"Start":"03:24.090 ","End":"03:26.000","Text":"That\u0027s the answer but wait,"},{"Start":"03:26.000 ","End":"03:28.970","Text":"we\u0027re not done because I said I\u0027m going to do it the other way in case you have"},{"Start":"03:28.970 ","End":"03:32.855","Text":"a more extensive expanded table of inverse transforms."},{"Start":"03:32.855 ","End":"03:37.580","Text":"I\u0027m going back to this point where we already decomposed the denominator."},{"Start":"03:37.580 ","End":"03:42.230","Text":"I\u0027m assuming now that we have a wider more expanded table of"},{"Start":"03:42.230 ","End":"03:44.540","Text":"inverse transforms and we have the formula for"},{"Start":"03:44.540 ","End":"03:48.725","Text":"this straightaway as this and now we can plug in right away."},{"Start":"03:48.725 ","End":"03:58.050","Text":"We can plug in that a is minus 3 and b is minus 2 because s plus 3,"},{"Start":"03:58.050 ","End":"03:59.310","Text":"that\u0027s minus minus 3."},{"Start":"03:59.310 ","End":"04:01.815","Text":"I\u0027ll spell it out, that\u0027s clear."},{"Start":"04:01.815 ","End":"04:05.465","Text":"Just a substitution gives us this,"},{"Start":"04:05.465 ","End":"04:07.720","Text":"and I\u0027ll just simplify it a bit,"},{"Start":"04:07.720 ","End":"04:11.570","Text":"minus and minus is a plus and this minus this is minus 1."},{"Start":"04:11.570 ","End":"04:15.755","Text":"Now all that\u0027s left is to bring the minus here."},{"Start":"04:15.755 ","End":"04:17.870","Text":"Make this a plus and this a minus,"},{"Start":"04:17.870 ","End":"04:19.140","Text":"and there we are,"},{"Start":"04:19.140 ","End":"04:21.510","Text":"it\u0027s the same result as we had before."},{"Start":"04:21.510 ","End":"04:24.470","Text":"That\u0027s just great and we\u0027re done."}],"ID":8015},{"Watched":false,"Name":"Exercise 14","Duration":"3m 5s","ChapterTopicVideoID":7943,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this exercise, we want to compute"},{"Start":"00:02.400 ","End":"00:07.020","Text":"the Inverse Laplace Transform of this rational function,"},{"Start":"00:07.020 ","End":"00:10.335","Text":"x^2 plus x minus 1 over s^3 minus s."},{"Start":"00:10.335 ","End":"00:15.000","Text":"The plan is to factorize the denominator and then you use partial fractions."},{"Start":"00:15.000 ","End":"00:17.010","Text":"Now the denominator is a cubic,"},{"Start":"00:17.010 ","End":"00:21.075","Text":"but it\u0027s easy to factorize it because we see that we can take s out the brackets,"},{"Start":"00:21.075 ","End":"00:23.205","Text":"that leaves us with x^2 minus 1."},{"Start":"00:23.205 ","End":"00:26.085","Text":"Now we can use the difference of squares on this."},{"Start":"00:26.085 ","End":"00:28.680","Text":"This part is s minus 1, s plus 1."},{"Start":"00:28.680 ","End":"00:31.110","Text":"Now our problem becomes this,"},{"Start":"00:31.110 ","End":"00:33.794","Text":"and now we can use partial fraction."},{"Start":"00:33.794 ","End":"00:38.420","Text":"Just note that the numerator here is a quadratic and the denominator is a cubic."},{"Start":"00:38.420 ","End":"00:40.145","Text":"We have a lower degree on top,"},{"Start":"00:40.145 ","End":"00:42.880","Text":"and that means we can use partial fractions."},{"Start":"00:42.880 ","End":"00:45.785","Text":"This is the general shape that we\u0027re going to convert it into,"},{"Start":"00:45.785 ","End":"00:48.200","Text":"and our task is to find the constants A, B,"},{"Start":"00:48.200 ","End":"00:52.315","Text":"and C. We multiply by the denominator,"},{"Start":"00:52.315 ","End":"00:54.075","Text":"and this is what we get."},{"Start":"00:54.075 ","End":"00:58.280","Text":"We get A times the 2 missing factors that are not s\u0027s and this,"},{"Start":"00:58.280 ","End":"01:01.645","Text":"b times this and this and so on."},{"Start":"01:01.645 ","End":"01:03.570","Text":"The way we find A,"},{"Start":"01:03.570 ","End":"01:05.485","Text":"B and C is by,"},{"Start":"01:05.485 ","End":"01:12.280","Text":"easiest here is to plug in judiciously the value of s. If I put s equals 1,"},{"Start":"01:12.280 ","End":"01:15.960","Text":"this term will disappear and so will this."},{"Start":"01:15.960 ","End":"01:21.110","Text":"If I put s is 0, this and this will disappear and so on and s"},{"Start":"01:21.110 ","End":"01:27.475","Text":"equals 0 gives us here and here as 0 and so we can get A to be 1."},{"Start":"01:27.475 ","End":"01:29.280","Text":"If we let s equals 1,"},{"Start":"01:29.280 ","End":"01:34.440","Text":"we get 1 equals 2B and B is 1/2 and if we let s be minus 1,"},{"Start":"01:34.440 ","End":"01:36.540","Text":"then this is 0 and this is 0,"},{"Start":"01:36.540 ","End":"01:40.515","Text":"so we get 2C is minus 1 and C is minus 1/2."},{"Start":"01:40.515 ","End":"01:46.755","Text":"I want to take these 3 values and substitute them here and I can now decompose this."},{"Start":"01:46.755 ","End":"01:49.890","Text":"What I get is this expression,"},{"Start":"01:49.890 ","End":"01:51.165","Text":"here\u0027s the 1,"},{"Start":"01:51.165 ","End":"01:52.590","Text":"here\u0027s the 1/2,"},{"Start":"01:52.590 ","End":"01:54.990","Text":"and here\u0027s the minus 1/2."},{"Start":"01:54.990 ","End":"02:00.050","Text":"Now, let\u0027s use linearity to break this up into pieces."},{"Start":"02:00.050 ","End":"02:04.550","Text":"This is the rule for linearity when we have 2 terms,"},{"Start":"02:04.550 ","End":"02:07.520","Text":"but it also works when you have 3 terms, just extended,"},{"Start":"02:07.520 ","End":"02:08.990","Text":"we have A, B, and C,"},{"Start":"02:08.990 ","End":"02:10.790","Text":"maybe G, H, I."},{"Start":"02:10.790 ","End":"02:12.290","Text":"Anyway, we break it up,"},{"Start":"02:12.290 ","End":"02:15.925","Text":"just means take each separately and take any constants out in front."},{"Start":"02:15.925 ","End":"02:18.455","Text":"Here are three separate inverse transforms,"},{"Start":"02:18.455 ","End":"02:21.560","Text":"and the constants are 1/2 and minus 1/2 in front."},{"Start":"02:21.560 ","End":"02:23.885","Text":"Now we can use the formula,"},{"Start":"02:23.885 ","End":"02:27.305","Text":"I mean, lookup in the inverse table,"},{"Start":"02:27.305 ","End":"02:31.490","Text":"and my table is actually 2 formulas for s minus a and s plus a and"},{"Start":"02:31.490 ","End":"02:36.395","Text":"really it\u0027s the same thing because I could put a to be positive or negative using these,"},{"Start":"02:36.395 ","End":"02:39.515","Text":"here this is S plus 0 or S minus 0."},{"Start":"02:39.515 ","End":"02:43.700","Text":"Actually, there\u0027s also a formula in the table for 1/S directly,"},{"Start":"02:43.700 ","End":"02:45.250","Text":"which is just 1,"},{"Start":"02:45.250 ","End":"02:47.790","Text":"and it also works if a is 0 here."},{"Start":"02:47.790 ","End":"02:51.065","Text":"In short, what we get is this,"},{"Start":"02:51.065 ","End":"02:54.475","Text":"this is the inverse of 1/s is 1."},{"Start":"02:54.475 ","End":"02:59.120","Text":"This we get by substituting a equals 1 in this formula."},{"Start":"02:59.120 ","End":"03:05.880","Text":"That\u0027s the 1/2 and minus 1/2 and here we put S equals 1 in this formula and that\u0027s it."}],"ID":8016},{"Watched":false,"Name":"Exercise 15","Duration":"3m 29s","ChapterTopicVideoID":7944,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.845","Text":"In this exercise, we want to find the Inverse Laplace Transform of this expression"},{"Start":"00:06.845 ","End":"00:14.010","Text":"and the general plan is to take this and first of all,"},{"Start":"00:14.010 ","End":"00:16.800","Text":"factorize the denominator and then use"},{"Start":"00:16.800 ","End":"00:20.835","Text":"partial fractions to help decompose this into simpler pieces."},{"Start":"00:20.835 ","End":"00:23.100","Text":"The first task is to take this denominator,"},{"Start":"00:23.100 ","End":"00:25.545","Text":"which is a cubic, and to break it up,"},{"Start":"00:25.545 ","End":"00:30.840","Text":"to factorize it completely into linear pieces and s minus something,"},{"Start":"00:30.840 ","End":"00:32.010","Text":"s minus something else,"},{"Start":"00:32.010 ","End":"00:33.540","Text":"s minus something else."},{"Start":"00:33.540 ","End":"00:35.640","Text":"The way we find s_1, s_2,"},{"Start":"00:35.640 ","End":"00:40.835","Text":"and s_3 would be to find the roots for the equation where this equals 0."},{"Start":"00:40.835 ","End":"00:44.660","Text":"Now this thing is a cubic equation and we don\u0027t know how to solve those,"},{"Start":"00:44.660 ","End":"00:46.250","Text":"not by formula anyway,"},{"Start":"00:46.250 ","End":"00:52.460","Text":"but there is a theorem that any whole number solution is going to be a factor of minus 6."},{"Start":"00:52.460 ","End":"00:56.720","Text":"If we write all the factors of minus 6 is actually 8 of them plus or minus 1,"},{"Start":"00:56.720 ","End":"00:58.145","Text":"2, 3, and 6."},{"Start":"00:58.145 ","End":"01:03.350","Text":"We just need to try them out systematically by plugging in and see which gives 0."},{"Start":"01:03.350 ","End":"01:06.890","Text":"I\u0027ll save you the trouble just tell you that minus 1, minus 2,"},{"Start":"01:06.890 ","End":"01:12.425","Text":"and 3 will do the trick and so this becomes s plus 1,"},{"Start":"01:12.425 ","End":"01:14.480","Text":"s plus 2, and s minus 3."},{"Start":"01:14.480 ","End":"01:17.775","Text":"Note the signs because it says minus whatever."},{"Start":"01:17.775 ","End":"01:21.950","Text":"Then plugging this in here with the factorized denominator,"},{"Start":"01:21.950 ","End":"01:26.065","Text":"we now have reduced our problem to this problem."},{"Start":"01:26.065 ","End":"01:27.860","Text":"Why is this easier?"},{"Start":"01:27.860 ","End":"01:30.410","Text":"Because here we can use partial fractions."},{"Start":"01:30.410 ","End":"01:32.750","Text":"Before I can use partial fractions I have to make sure that"},{"Start":"01:32.750 ","End":"01:35.375","Text":"the degree on top is less than the degree on the bottom."},{"Start":"01:35.375 ","End":"01:36.590","Text":"Here the degree is 3,"},{"Start":"01:36.590 ","End":"01:38.725","Text":"here the degree is 2, so we\u0027re all right."},{"Start":"01:38.725 ","End":"01:42.785","Text":"Then we have separate factors we just put a constant above each of them,"},{"Start":"01:42.785 ","End":"01:45.470","Text":"like so and our task is to find A,"},{"Start":"01:45.470 ","End":"01:50.938","Text":"B, and C. So we multiply both sides by this denominator,"},{"Start":"01:50.938 ","End":"01:52.625","Text":"and this is what we get."},{"Start":"01:52.625 ","End":"01:57.920","Text":"We\u0027re going to use our usual trick of substituting judiciously values of s. In fact,"},{"Start":"01:57.920 ","End":"02:02.210","Text":"we\u0027re going to try substituting minus 1 the first time,"},{"Start":"02:02.210 ","End":"02:04.625","Text":"and that will make this and this 0."},{"Start":"02:04.625 ","End":"02:10.070","Text":"Next, we\u0027ll substitute minus 2 and that would make this and this 0."},{"Start":"02:10.070 ","End":"02:12.020","Text":"Then we\u0027ll substitute 3,"},{"Start":"02:12.020 ","End":"02:14.650","Text":"I mean these are just the 3 roots that we found."},{"Start":"02:14.650 ","End":"02:16.400","Text":"I\u0027ll give them all 3 at once."},{"Start":"02:16.400 ","End":"02:17.930","Text":"If you just follow these computations,"},{"Start":"02:17.930 ","End":"02:22.180","Text":"you\u0027ll see that each time we can find 1 of the variables, here we find A."},{"Start":"02:22.180 ","End":"02:23.275","Text":"This one gives us B."},{"Start":"02:23.275 ","End":"02:27.200","Text":"This one gives us C and now we just have to plug these into"},{"Start":"02:27.200 ","End":"02:31.890","Text":"here and so we get this here the constants A,"},{"Start":"02:31.890 ","End":"02:34.575","Text":"B, C just happened to be 1, 2, 3."},{"Start":"02:34.575 ","End":"02:40.175","Text":"Now we want to split this up using the linearity of the inverse transform."},{"Start":"02:40.175 ","End":"02:42.890","Text":"I don\u0027t know why this thing came in again,"},{"Start":"02:42.890 ","End":"02:45.140","Text":"I\u0027ll just ignore the duplicate."},{"Start":"02:45.140 ","End":"02:48.320","Text":"I meant to show you the rule for the linearity of"},{"Start":"02:48.320 ","End":"02:52.700","Text":"the inverse transform and although here it appears with just 2 terms,"},{"Start":"02:52.700 ","End":"02:54.170","Text":"it works for 3 terms,"},{"Start":"02:54.170 ","End":"02:57.020","Text":"also something plus something plus something it means we"},{"Start":"02:57.020 ","End":"03:00.320","Text":"just break up into 3 pieces and pull constants out"},{"Start":"03:00.320 ","End":"03:02.540","Text":"and from here we end up with this note that the"},{"Start":"03:02.540 ","End":"03:04.700","Text":"2 and the 3 we\u0027ve pulled out and it\u0027s broken up into"},{"Start":"03:04.700 ","End":"03:10.690","Text":"3 separate pieces and now we look in the table of inverse transforms for these 3."},{"Start":"03:10.690 ","End":"03:12.825","Text":"We find these 2 rules,"},{"Start":"03:12.825 ","End":"03:14.460","Text":"one with an s minus a,"},{"Start":"03:14.460 ","End":"03:19.835","Text":"one with an s plus a. I guess we\u0027re going to use this rule here and here with"},{"Start":"03:19.835 ","End":"03:25.505","Text":"a being 1 and 2 and this rule here with a is 3 and if we do that,"},{"Start":"03:25.505 ","End":"03:29.910","Text":"this is what we get and that\u0027s the answer so we\u0027re done."}],"ID":8017},{"Watched":false,"Name":"Exercise 16","Duration":"6m 29s","ChapterTopicVideoID":7948,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we want to compute"},{"Start":"00:02.310 ","End":"00:09.945","Text":"the Inverse Laplace Transform of this,10s over s^4 minus 13s^2 plus 36."},{"Start":"00:09.945 ","End":"00:13.170","Text":"There\u0027s nothing quite like this in the table of"},{"Start":"00:13.170 ","End":"00:16.980","Text":"inverse transform so we\u0027re going to have to do a bit of algebra on this."},{"Start":"00:16.980 ","End":"00:21.465","Text":"What I\u0027d like to do is to factorize the denominator completely."},{"Start":"00:21.465 ","End":"00:23.955","Text":"It\u0027s a fourth degree polynomial in s,"},{"Start":"00:23.955 ","End":"00:26.715","Text":"and it should have 4 roots,"},{"Start":"00:26.715 ","End":"00:32.010","Text":"and say s_1 through s_4 and then we could write it like this"},{"Start":"00:32.010 ","End":"00:38.295","Text":"and s_1-s_4 are the solutions or roots of this equation."},{"Start":"00:38.295 ","End":"00:42.650","Text":"Now it\u0027s a fourth degree equation but notice that it only has even powers,"},{"Start":"00:42.650 ","End":"00:45.020","Text":"the s^3 is missing and the s is missing,"},{"Start":"00:45.020 ","End":"00:50.494","Text":"so there\u0027s a standard trick here to substitute s^2 equals something."},{"Start":"00:50.494 ","End":"00:52.835","Text":"I\u0027ll use the letter t maybe not the best"},{"Start":"00:52.835 ","End":"00:55.790","Text":"letter because the inverse transform is going to be with t,"},{"Start":"00:55.790 ","End":"00:59.585","Text":"it doesn\u0027t matter we won\u0027t get confused and so if we do the substitution,"},{"Start":"00:59.585 ","End":"01:03.770","Text":"we\u0027ll get t^2 minus 13t plus 36 equals"},{"Start":"01:03.770 ","End":"01:09.245","Text":"0 and that\u0027s a regular quadratic equation and if we solve this equation,"},{"Start":"01:09.245 ","End":"01:12.590","Text":"then we get 2 roots, 9 and 4."},{"Start":"01:12.590 ","End":"01:16.940","Text":"But we don\u0027t want t we want s and t is s^2,"},{"Start":"01:16.940 ","End":"01:20.510","Text":"so s^2 is 9 or s^2 is 4."},{"Start":"01:20.510 ","End":"01:25.010","Text":"This one gives us plus or minus 3 for s and this one gives"},{"Start":"01:25.010 ","End":"01:29.360","Text":"us plus or minus 2 and if I just write them all out, this particular order,"},{"Start":"01:29.360 ","End":"01:33.505","Text":"let\u0027s say, then I\u0027ve got all for s\u0027s and that means I can"},{"Start":"01:33.505 ","End":"01:39.515","Text":"factorize this fourth degree polynomial into 4 linear factors."},{"Start":"01:39.515 ","End":"01:41.155","Text":"Then our problem,"},{"Start":"01:41.155 ","End":"01:44.825","Text":"we can rewrite as this and this is much better"},{"Start":"01:44.825 ","End":"01:48.920","Text":"because I can do partial fractions on this,"},{"Start":"01:48.920 ","End":"01:51.125","Text":"if you remember your partial fractions."},{"Start":"01:51.125 ","End":"01:54.680","Text":"What we\u0027re going to do is rewrite this rational function,"},{"Start":"01:54.680 ","End":"01:59.750","Text":"it is in the following form I take all of these linear factors"},{"Start":"01:59.750 ","End":"02:05.120","Text":"here and I put constants above them and my job is now to find A,"},{"Start":"02:05.120 ","End":"02:09.005","Text":"B, C, and D using the method of partial fractions."},{"Start":"02:09.005 ","End":"02:10.820","Text":"Now we multiply everything by"},{"Start":"02:10.820 ","End":"02:16.160","Text":"this denominator and it\u0027s a bit messy but really it\u0027s not too"},{"Start":"02:16.160 ","End":"02:23.570","Text":"bad because each time I can make a substitution to make 3 out of the 4 terms disappear."},{"Start":"02:23.570 ","End":"02:25.820","Text":"I\u0027m going to substitute minus 3,"},{"Start":"02:25.820 ","End":"02:28.685","Text":"3 minus 2 and 2 respectively."},{"Start":"02:28.685 ","End":"02:31.480","Text":"If we substitute s is minus 3,"},{"Start":"02:31.480 ","End":"02:37.447","Text":"then everything but the first one becomes 0 because I\u0027ve got an s plus 3, s plus 3,"},{"Start":"02:37.447 ","End":"02:44.630","Text":"s plus 3 and so we get here minus 30 and here just the first term A times,"},{"Start":"02:44.630 ","End":"02:48.080","Text":"well, if you substitute minus 3 you get minus 30 and that"},{"Start":"02:48.080 ","End":"02:51.900","Text":"gives us A=1 and therefore we let s=3."},{"Start":"02:51.900 ","End":"02:55.905","Text":"I\u0027ll leave you to verify this, we get B=1."},{"Start":"02:55.905 ","End":"02:58.500","Text":"If we put s is minus 2,"},{"Start":"02:58.500 ","End":"02:59.970","Text":"then we get the value of C,"},{"Start":"02:59.970 ","End":"03:03.825","Text":"which is minus 1 and then we still need D,"},{"Start":"03:03.825 ","End":"03:10.230","Text":"which we get by substituting s=2 in this and then we\u0027ve got D so now we have A,"},{"Start":"03:10.230 ","End":"03:14.085","Text":"B, C, and D, and we need to substitute them."},{"Start":"03:14.085 ","End":"03:18.815","Text":"Our expression or function that we want to find the inverse transform of,"},{"Start":"03:18.815 ","End":"03:21.440","Text":"is like this, is the A, the B,"},{"Start":"03:21.440 ","End":"03:22.700","Text":"the C is minus 1,"},{"Start":"03:22.700 ","End":"03:24.460","Text":"the D is minus 1."},{"Start":"03:24.460 ","End":"03:28.410","Text":"Now the Inverse Laplace Transform is linear."},{"Start":"03:28.410 ","End":"03:31.790","Text":"The linear property is this if it\u0027s"},{"Start":"03:31.790 ","End":"03:35.420","Text":"2 terms but it also works for 3 and 4 or any number of terms,"},{"Start":"03:35.420 ","End":"03:37.730","Text":"here we have 4 terms it\u0027s the same idea."},{"Start":"03:37.730 ","End":"03:42.222","Text":"We break it up and pull constants out and this is what we get,"},{"Start":"03:42.222 ","End":"03:47.172","Text":"we get 4 separate inverse transforms and here are the constants,"},{"Start":"03:47.172 ","End":"03:50.480","Text":"well the 1 and 1 we don\u0027t need to do anything but the minus 1 is here,"},{"Start":"03:50.480 ","End":"03:55.340","Text":"minus is here and all these 4 are easy,"},{"Start":"03:55.340 ","End":"03:59.945","Text":"there\u0027s straightforward lookup in the inverse transform table."},{"Start":"03:59.945 ","End":"04:03.680","Text":"Well, we have these 2 general rules with a general A,"},{"Start":"04:03.680 ","End":"04:08.295","Text":"and we can let A here be 3 and 2 and for this,"},{"Start":"04:08.295 ","End":"04:10.800","Text":"we can set A be 3 and 2 also,"},{"Start":"04:10.800 ","End":"04:13.700","Text":"and if we do that,"},{"Start":"04:13.700 ","End":"04:16.730","Text":"we get the following solution."},{"Start":"04:16.730 ","End":"04:21.875","Text":"Here\u0027s the minus 3 here I plugged in 3 in"},{"Start":"04:21.875 ","End":"04:28.385","Text":"this formula and then minus 2 here and then plus 2 here and this is what we get."},{"Start":"04:28.385 ","End":"04:34.220","Text":"It is the answer, but I want to also show you another method but this other method"},{"Start":"04:34.220 ","End":"04:35.510","Text":"assumes that you have"},{"Start":"04:35.510 ","End":"04:40.925","Text":"a more extensive table of Inverse Laplace Transform so you\u0027ll see in a moment."},{"Start":"04:40.925 ","End":"04:43.820","Text":"Well, I take the 10 out using linearity,"},{"Start":"04:43.820 ","End":"04:46.535","Text":"that\u0027s okay and what I do is I combine these 2,"},{"Start":"04:46.535 ","End":"04:52.280","Text":"and I combine these 2 and this is what we get with a difference of squares formula."},{"Start":"04:52.280 ","End":"04:56.225","Text":"Actually, we could have done this directly because previously we had"},{"Start":"04:56.225 ","End":"05:00.760","Text":"t was s^2 and that came out to be either"},{"Start":"05:00.760 ","End":"05:05.430","Text":"9 or 4 that was 3^2 or"},{"Start":"05:05.430 ","End":"05:11.505","Text":"2^2 and we could have done this directly and not have to break it up so much, anyway."},{"Start":"05:11.505 ","End":"05:14.929","Text":"Now if you have a good table of inverse transforms,"},{"Start":"05:14.929 ","End":"05:19.610","Text":"then you\u0027ll find this formula and this formula is very much like"},{"Start":"05:19.610 ","End":"05:24.805","Text":"this just a is 3 and b is 2 it\u0027s practically almost done."},{"Start":"05:24.805 ","End":"05:28.940","Text":"I have this and all I have to do is plug in a=3,"},{"Start":"05:28.940 ","End":"05:32.685","Text":"b=2 and this is what we get now look,"},{"Start":"05:32.685 ","End":"05:35.325","Text":"9 minus 4 is 5,"},{"Start":"05:35.325 ","End":"05:41.690","Text":"5 into 10 goes twice and this is the answer and they might say,"},{"Start":"05:41.690 ","End":"05:44.780","Text":"wait a minute, this is not the same as what we got before."},{"Start":"05:44.780 ","End":"05:51.165","Text":"Well, it doesn\u0027t look like it but if you remember that cosh,"},{"Start":"05:51.165 ","End":"05:54.845","Text":"hyperbolic cosine, if you remember its definition,"},{"Start":"05:54.845 ","End":"05:56.648","Text":"then this is,"},{"Start":"05:56.648 ","End":"05:57.740","Text":"I\u0027ll write it down,"},{"Start":"05:57.740 ","End":"05:59.810","Text":"I\u0027ll show you, this is twice,"},{"Start":"05:59.810 ","End":"06:07.370","Text":"this is e^3t plus e to the minus 3t/2 and this"},{"Start":"06:07.370 ","End":"06:15.180","Text":"is e^2t plus e to the minus 2t also over 2,"},{"Start":"06:15.180 ","End":"06:19.740","Text":"it\u0027s always over 2 it\u0027s nothing to do with this 2 and then this 2 goes"},{"Start":"06:19.740 ","End":"06:24.785","Text":"with this and this and if you write this plus this minus this minus this,"},{"Start":"06:24.785 ","End":"06:28.280","Text":"then you\u0027ll see that it is the same as what we had before."},{"Start":"06:28.280 ","End":"06:30.450","Text":"So we\u0027re okay."}],"ID":8018},{"Watched":false,"Name":"Exercise 17","Duration":"2m 55s","ChapterTopicVideoID":7949,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.425","Text":"Here, we want to compute the Inverse Laplace Transform of 8s over x minus 2^2 s plus 2."},{"Start":"00:07.425 ","End":"00:10.410","Text":"What we\u0027re going to do is to break up"},{"Start":"00:10.410 ","End":"00:14.010","Text":"this rational expression in s into partial fractions."},{"Start":"00:14.010 ","End":"00:17.190","Text":"Then we\u0027ll be able to use the lookup table because this as is,"},{"Start":"00:17.190 ","End":"00:19.600","Text":"it\u0027s not in the table of inverse transforms."},{"Start":"00:19.600 ","End":"00:25.125","Text":"Now the denominator is already factorized so that\u0027s easier."},{"Start":"00:25.125 ","End":"00:27.950","Text":"Good to remember that when we have a double factor,"},{"Start":"00:27.950 ","End":"00:36.105","Text":"like a double root here then we have A over s minus 2 and also s minus 2^2."},{"Start":"00:36.105 ","End":"00:38.160","Text":"We have to represent all exponents,"},{"Start":"00:38.160 ","End":"00:44.710","Text":"and for C is just C over s plus 2 multiplied by the common denominator here."},{"Start":"00:44.710 ","End":"00:47.130","Text":"This is what we get."},{"Start":"00:47.130 ","End":"00:49.500","Text":"To make it easier to find A, B, and C,"},{"Start":"00:49.500 ","End":"00:52.835","Text":"we can substitute any value of s we want."},{"Start":"00:52.835 ","End":"00:56.950","Text":"Now 2 good values would be 2 and minus 2."},{"Start":"00:56.950 ","End":"00:58.200","Text":"We substitute 2,"},{"Start":"00:58.200 ","End":"01:03.615","Text":"this is 0 and this is 0 and simple computation gives us that B is 4."},{"Start":"01:03.615 ","End":"01:05.775","Text":"If s is minus 2,"},{"Start":"01:05.775 ","End":"01:13.250","Text":"then this and this becomes 0 and we just have this which gives us what C is,"},{"Start":"01:13.250 ","End":"01:15.300","Text":"turns out to be minus 1."},{"Start":"01:15.300 ","End":"01:19.460","Text":"This is the usual thing we do to make one or more of the factors 0."},{"Start":"01:19.460 ","End":"01:21.950","Text":"Now, besides 2 and minus 2,"},{"Start":"01:21.950 ","End":"01:24.470","Text":"I don\u0027t know what else, so I\u0027ll just choose anything convenient."},{"Start":"01:24.470 ","End":"01:26.720","Text":"Now, 0 is often a value to compute with,"},{"Start":"01:26.720 ","End":"01:28.445","Text":"so let\u0027s try x equals 0."},{"Start":"01:28.445 ","End":"01:30.235","Text":"That gives us this."},{"Start":"01:30.235 ","End":"01:36.350","Text":"But remember we already have B and C. B is 4 and C is minus 1."},{"Start":"01:36.350 ","End":"01:38.390","Text":"If you substitute that in,"},{"Start":"01:38.390 ","End":"01:40.355","Text":"then you\u0027ll get that a is 1."},{"Start":"01:40.355 ","End":"01:43.945","Text":"We have A, B, and C, and we put them here, here and here."},{"Start":"01:43.945 ","End":"01:49.370","Text":"Now our problem becomes to find the inverse transform of"},{"Start":"01:49.370 ","End":"01:55.940","Text":"this sum and we\u0027re going to use the linearity property of the inverse transform,"},{"Start":"01:55.940 ","End":"01:58.640","Text":"and just for reference, I\u0027ve brought it here again."},{"Start":"01:58.640 ","End":"02:02.495","Text":"This is the linearity for 2 terms,"},{"Start":"02:02.495 ","End":"02:05.305","Text":"but it also works for 3 terms in a similar way."},{"Start":"02:05.305 ","End":"02:07.760","Text":"Basically it means we break it up into bits,"},{"Start":"02:07.760 ","End":"02:09.200","Text":"we apply L minus 1 to this,"},{"Start":"02:09.200 ","End":"02:12.050","Text":"this and this, and here and here we take the constant out."},{"Start":"02:12.050 ","End":"02:13.120","Text":"Here we take the 4 out,"},{"Start":"02:13.120 ","End":"02:15.078","Text":"here we take the minus 1 out,"},{"Start":"02:15.078 ","End":"02:19.549","Text":"and now we need to find the inverse transforms for these 3 pieces,"},{"Start":"02:19.549 ","End":"02:24.370","Text":"and we look in the table and we find 3 formulas that are going to help us."},{"Start":"02:24.370 ","End":"02:27.620","Text":"We have 1 over s minus a and that\u0027ll do us for this,"},{"Start":"02:27.620 ","End":"02:29.575","Text":"if I let A equals 2."},{"Start":"02:29.575 ","End":"02:33.410","Text":"Let\u0027s just say this one goes to this one, A is 2."},{"Start":"02:33.410 ","End":"02:39.193","Text":"This one will go with this one also if I let A=2,"},{"Start":"02:39.193 ","End":"02:45.750","Text":"and this one will go with this one if I let A=2."},{"Start":"02:45.750 ","End":"02:47.865","Text":"It\u0027s a different A in each of them."},{"Start":"02:47.865 ","End":"02:55.950","Text":"Plug all these values in and it\u0027s easy to see that this is what we get and we are done."}],"ID":8019},{"Watched":false,"Name":"Exercise 18","Duration":"2m 40s","ChapterTopicVideoID":7950,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.455","Text":"Here we have to compute the Inverse Laplace Transform of 5 minus s over s^3 plus s^2."},{"Start":"00:06.455 ","End":"00:08.760","Text":"We\u0027ve already done enough of these to recognize that"},{"Start":"00:08.760 ","End":"00:11.220","Text":"this is the case for partial fractions."},{"Start":"00:11.220 ","End":"00:14.970","Text":"But the first step will be to factorize the denominator here."},{"Start":"00:14.970 ","End":"00:18.420","Text":"Clearly, this is s^2 times s plus 1,"},{"Start":"00:18.420 ","End":"00:23.705","Text":"so s=0 is a double root and minus 1 is a single root."},{"Start":"00:23.705 ","End":"00:28.560","Text":"If I rephrase that, s is a double factor and s plus 1 is a single factor."},{"Start":"00:28.560 ","End":"00:29.955","Text":"For the s part,"},{"Start":"00:29.955 ","End":"00:33.650","Text":"we have to take s and s^2 in our partial fraction."},{"Start":"00:33.650 ","End":"00:35.035","Text":"But for the s plus 1,"},{"Start":"00:35.035 ","End":"00:37.440","Text":"it\u0027s sufficient as it is."},{"Start":"00:37.440 ","End":"00:44.640","Text":"We have A, B, and C. Next thing we do is multiply by this denominator."},{"Start":"00:44.640 ","End":"00:46.619","Text":"That gives us this expression."},{"Start":"00:46.619 ","End":"00:47.970","Text":"You know what\u0027s coming next."},{"Start":"00:47.970 ","End":"00:51.740","Text":"We\u0027re going to substitute 3 different values of s here."},{"Start":"00:51.740 ","End":"00:53.060","Text":"The easiest, of course,"},{"Start":"00:53.060 ","End":"00:56.115","Text":"is what makes 1 or more of these factors 0."},{"Start":"00:56.115 ","End":"00:59.430","Text":"I would let x=0 and s equal minus 1,"},{"Start":"00:59.430 ","End":"01:01.005","Text":"and then see where we are."},{"Start":"01:01.005 ","End":"01:03.620","Text":"For s=0, this is the computation."},{"Start":"01:03.620 ","End":"01:04.730","Text":"I\u0027ll leave you to follow it."},{"Start":"01:04.730 ","End":"01:06.130","Text":"We get B=5."},{"Start":"01:06.130 ","End":"01:08.120","Text":"For s is minus 1."},{"Start":"01:08.120 ","End":"01:10.110","Text":"As before, we get 2 zeros."},{"Start":"01:10.110 ","End":"01:11.240","Text":"Here, we get 2 zeros,"},{"Start":"01:11.240 ","End":"01:14.225","Text":"and so nicely, C comes out to be 6."},{"Start":"01:14.225 ","End":"01:17.650","Text":"Now a third value is anything convenient."},{"Start":"01:17.650 ","End":"01:19.130","Text":"I usually say 0,"},{"Start":"01:19.130 ","End":"01:21.110","Text":"but we\u0027ve used 0 up already."},{"Start":"01:21.110 ","End":"01:24.290","Text":"Lets let s=1. It\u0027s a typo."},{"Start":"01:24.290 ","End":"01:29.615","Text":"Let\u0027s make that s. Then we get this equation."},{"Start":"01:29.615 ","End":"01:35.300","Text":"But we already know that B is 5 and we know that C is 6,"},{"Start":"01:35.300 ","End":"01:36.800","Text":"so the only unknown here is A,"},{"Start":"01:36.800 ","End":"01:39.650","Text":"and if you do the algebra or the math,"},{"Start":"01:39.650 ","End":"01:41.315","Text":"you get A is minus 6."},{"Start":"01:41.315 ","End":"01:47.120","Text":"We have A, B, and C. Plug them into here and our problem reduces to this."},{"Start":"01:47.120 ","End":"01:51.110","Text":"Next, we apply linearity and break this up."},{"Start":"01:51.110 ","End":"01:53.960","Text":"I just quoted the linearity rule."},{"Start":"01:53.960 ","End":"01:59.120","Text":"This is the linearity rule for 2 terms,"},{"Start":"01:59.120 ","End":"02:02.830","Text":"but it works for 3 also or any number."},{"Start":"02:02.830 ","End":"02:05.045","Text":"This breaks up into this."},{"Start":"02:05.045 ","End":"02:06.470","Text":"Notice the constants come out in"},{"Start":"02:06.470 ","End":"02:08.780","Text":"front and we have 2 pluses."},{"Start":"02:08.780 ","End":"02:08.781","Text":"This is the expression we get."},{"Start":"02:08.781 ","End":"02:09.782","Text":"Then we go to the look-up table of inverse Laplace transforms."},{"Start":"02:24.500 ","End":"02:25.500","Text":"Couple of them, we have immediately. We have this one right here."},{"Start":"02:25.500 ","End":"02:25.501","Text":"We have this one right here."},{"Start":"02:25.501 ","End":"02:26.990","Text":"This one, well, we have very close,"},{"Start":"02:26.990 ","End":"02:28.295","Text":"we have at the general a."},{"Start":"02:28.295 ","End":"02:32.105","Text":"All I have to do is let a=2 in this one."},{"Start":"02:32.105 ","End":"02:34.220","Text":"Just piece this together,"},{"Start":"02:34.220 ","End":"02:36.120","Text":"this is what we have, 6 times 1,"},{"Start":"02:36.120 ","End":"02:37.230","Text":"5 times t,"},{"Start":"02:37.230 ","End":"02:41.620","Text":"and 6e to the minus 2t. We\u0027re done."}],"ID":8020},{"Watched":false,"Name":"Exercise 19","Duration":"2m 51s","ChapterTopicVideoID":7951,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.670","Text":"Here, we\u0027re asked to compute the Inverse Laplace Transform of this expression."},{"Start":"00:05.670 ","End":"00:07.485","Text":"It\u0027s a rational expression,"},{"Start":"00:07.485 ","End":"00:11.160","Text":"and the degree in the numerator is less than that in the denominator."},{"Start":"00:11.160 ","End":"00:15.150","Text":"We know that this exercise works with partial fractions."},{"Start":"00:15.150 ","End":"00:17.625","Text":"But the first step in the partial fractions is to"},{"Start":"00:17.625 ","End":"00:20.280","Text":"decompose the denominator, to factorize it."},{"Start":"00:20.280 ","End":"00:23.190","Text":"The first thing to do is take s outside the brackets."},{"Start":"00:23.190 ","End":"00:27.525","Text":"If you look at this,1 of those special binomial expansions, you know what I mean,"},{"Start":"00:27.525 ","End":"00:32.280","Text":"a^2 plus 2ab plus b^2 equals a plus b squared,"},{"Start":"00:32.280 ","End":"00:33.510","Text":"and a will be s,"},{"Start":"00:33.510 ","End":"00:34.905","Text":"and b will be 3,"},{"Start":"00:34.905 ","End":"00:37.365","Text":"and so this is our factorization,"},{"Start":"00:37.365 ","End":"00:38.925","Text":"s is a single factor,"},{"Start":"00:38.925 ","End":"00:41.565","Text":"but that\u0027s plus 3 is a double factor."},{"Start":"00:41.565 ","End":"00:44.000","Text":"When we do the partial fractions,"},{"Start":"00:44.000 ","End":"00:48.535","Text":"we have to take this squared and to the power of 1,"},{"Start":"00:48.535 ","End":"00:50.850","Text":"so a/s as usual,"},{"Start":"00:50.850 ","End":"00:53.310","Text":"and the s plus 3^2 gives us x plus 3^2,"},{"Start":"00:53.310 ","End":"00:54.540","Text":"and s plus 3,"},{"Start":"00:54.540 ","End":"00:57.030","Text":"2 different constants here."},{"Start":"00:57.030 ","End":"00:59.070","Text":"We do this the usual way,"},{"Start":"00:59.070 ","End":"01:05.090","Text":"we multiply both sides by this denominator to get rid of denominators,"},{"Start":"01:05.090 ","End":"01:06.965","Text":"and this is what we get."},{"Start":"01:06.965 ","End":"01:08.540","Text":"Now we try and find a, b,"},{"Start":"01:08.540 ","End":"01:12.290","Text":"and c by plugging in suitable values of s. For one thing we could"},{"Start":"01:12.290 ","End":"01:16.435","Text":"do is substitute s equals minus 3."},{"Start":"01:16.435 ","End":"01:19.755","Text":"Start with 0, that makes this and this 0."},{"Start":"01:19.755 ","End":"01:22.295","Text":"If you follow this, we just get a equals 4."},{"Start":"01:22.295 ","End":"01:23.570","Text":"If s is minus 3,"},{"Start":"01:23.570 ","End":"01:25.490","Text":"then this becomes 0 and this becomes 0."},{"Start":"01:25.490 ","End":"01:27.155","Text":"Then where we can find c,"},{"Start":"01:27.155 ","End":"01:29.045","Text":"and it comes out to be minus 3,"},{"Start":"01:29.045 ","End":"01:30.670","Text":"9 over minus 3."},{"Start":"01:30.670 ","End":"01:33.260","Text":"Lastly, we just pick any value."},{"Start":"01:33.260 ","End":"01:34.700","Text":"I\u0027ll choose s=1,"},{"Start":"01:34.700 ","End":"01:38.120","Text":"but you could have picked anything except 0 and minus 3."},{"Start":"01:38.120 ","End":"01:39.800","Text":"That gives us this,"},{"Start":"01:39.800 ","End":"01:45.540","Text":"but we already have that a is 4 and C is minus 3."},{"Start":"01:45.540 ","End":"01:48.630","Text":"If you do the math B comes out minus 4,"},{"Start":"01:48.630 ","End":"01:51.475","Text":"and now I can plug in these values into here."},{"Start":"01:51.475 ","End":"01:55.445","Text":"The original problem breaks down into an easier problem like this,"},{"Start":"01:55.445 ","End":"02:00.605","Text":"which we can further break down using the linearity of the inverse transform."},{"Start":"02:00.605 ","End":"02:06.005","Text":"Just for formality, write the rule for linearity but should know it by now."},{"Start":"02:06.005 ","End":"02:09.950","Text":"We break this up into 3 bits and pull out the constants,"},{"Start":"02:09.950 ","End":"02:11.510","Text":"and we end up with this,"},{"Start":"02:11.510 ","End":"02:13.940","Text":"and we need 3 inverse transforms."},{"Start":"02:13.940 ","End":"02:15.155","Text":"This, this, and this,"},{"Start":"02:15.155 ","End":"02:17.390","Text":"this 1 we have exactly in the table,"},{"Start":"02:17.390 ","End":"02:20.570","Text":"this and this we have close, when I say close,"},{"Start":"02:20.570 ","End":"02:22.070","Text":"I mean we have the general a,"},{"Start":"02:22.070 ","End":"02:24.140","Text":"is like I could let a equals 3 here,"},{"Start":"02:24.140 ","End":"02:27.730","Text":"and get this and also a equals 3 here and get this,"},{"Start":"02:27.730 ","End":"02:29.260","Text":"and this 1 as is."},{"Start":"02:29.260 ","End":"02:31.650","Text":"If I do all that, I\u0027ll get this."},{"Start":"02:31.650 ","End":"02:34.080","Text":"We get 4 times the 1."},{"Start":"02:34.080 ","End":"02:38.375","Text":"From here we have e to the minus 3t,"},{"Start":"02:38.375 ","End":"02:40.325","Text":"but there\u0027s also a minus 4,"},{"Start":"02:40.325 ","End":"02:43.250","Text":"and here when a is 3,"},{"Start":"02:43.250 ","End":"02:46.040","Text":"we get te to the minus 3t,"},{"Start":"02:46.040 ","End":"02:47.870","Text":"but already was a minus 3."},{"Start":"02:47.870 ","End":"02:51.750","Text":"In short, this is the answer, and we\u0027re done."}],"ID":8021},{"Watched":false,"Name":"Exercise 20","Duration":"3m 48s","ChapterTopicVideoID":7952,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we want to compute the Inverse Laplace Transform of this expression."},{"Start":"00:05.730 ","End":"00:09.780","Text":"Notice that the denominator is actually of degree 4,"},{"Start":"00:09.780 ","End":"00:13.200","Text":"so we\u0027ll have to work a bit harder than usual."},{"Start":"00:13.200 ","End":"00:16.380","Text":"You probably guessed that we\u0027re going to use partial fractions,"},{"Start":"00:16.380 ","End":"00:18.180","Text":"and that\u0027s the method I\u0027d recommend."},{"Start":"00:18.180 ","End":"00:21.990","Text":"But the first thing to do would be to factorize the denominator."},{"Start":"00:21.990 ","End":"00:24.510","Text":"It\u0027s partly factored but not completely."},{"Start":"00:24.510 ","End":"00:27.435","Text":"You can probably see immediately that this is"},{"Start":"00:27.435 ","End":"00:30.300","Text":"s minus 1^2 and this is s minus 2^2, and if not,"},{"Start":"00:30.300 ","End":"00:38.235","Text":"I\u0027ll just remind you that a minus b^2 is a^2 minus 2ab plus b^2,"},{"Start":"00:38.235 ","End":"00:40.560","Text":"and that should help you to see this."},{"Start":"00:40.560 ","End":"00:42.690","Text":"Now, we have 4 factors."},{"Start":"00:42.690 ","End":"00:45.735","Text":"We have s minus 1 twice and s minus 2 twice,"},{"Start":"00:45.735 ","End":"00:48.065","Text":"so the general shape of the partial fraction,"},{"Start":"00:48.065 ","End":"00:50.690","Text":"I\u0027ll have to take s minus 1 and s minus 1^2 and"},{"Start":"00:50.690 ","End":"00:55.015","Text":"also s minus 2 an s minus 2^2 and 4 different constants."},{"Start":"00:55.015 ","End":"00:57.575","Text":"It\u0027ll be a bit more work than usual."},{"Start":"00:57.575 ","End":"00:59.720","Text":"Still, we started off by multiplying"},{"Start":"00:59.720 ","End":"01:02.680","Text":"both sides by this and getting rid of all denominator."},{"Start":"01:02.680 ","End":"01:04.695","Text":"That will give us this."},{"Start":"01:04.695 ","End":"01:10.890","Text":"Now, the next thing is to substitute 4 different values of s to get 4 equations,"},{"Start":"01:10.890 ","End":"01:13.700","Text":"but we want to make smart choices for s,"},{"Start":"01:13.700 ","End":"01:15.530","Text":"though any foreign principal will do."},{"Start":"01:15.530 ","End":"01:18.230","Text":"But look if we got x=1 or x=2,"},{"Start":"01:18.230 ","End":"01:20.945","Text":"a lot of these factors will be 0 and it\u0027ll be easier."},{"Start":"01:20.945 ","End":"01:22.765","Text":"For let s=1,"},{"Start":"01:22.765 ","End":"01:25.055","Text":"everything, but this is 0."},{"Start":"01:25.055 ","End":"01:27.635","Text":"Anyway, you can follow this and I get B is 1."},{"Start":"01:27.635 ","End":"01:29.330","Text":"If s is 2,"},{"Start":"01:29.330 ","End":"01:34.175","Text":"but the last factor come out 0 and we get D. Now,"},{"Start":"01:34.175 ","End":"01:36.004","Text":"what do we do for 2 other values?"},{"Start":"01:36.004 ","End":"01:40.245","Text":"Well, in truth, you could substitute any 2 values other than 1 and 2,"},{"Start":"01:40.245 ","End":"01:44.490","Text":"but here again, convenience 0 is always easy to substitute,"},{"Start":"01:44.490 ","End":"01:46.310","Text":"and if we do that we get this."},{"Start":"01:46.310 ","End":"01:48.350","Text":"But out of these 4 unknowns,"},{"Start":"01:48.350 ","End":"01:52.250","Text":"we know B is 1 and we know that D is 1,"},{"Start":"01:52.250 ","End":"01:55.850","Text":"so we\u0027ve got an equation in A and C. We don\u0027t automatically"},{"Start":"01:55.850 ","End":"01:59.510","Text":"or immediately get one of these other 2 constants."},{"Start":"01:59.510 ","End":"02:01.235","Text":"If we do something similar,"},{"Start":"02:01.235 ","End":"02:03.380","Text":"take s=3 or anything,"},{"Start":"02:03.380 ","End":"02:04.940","Text":"take minus 1, whatever."},{"Start":"02:04.940 ","End":"02:06.485","Text":"Anyway, I\u0027m taking 3."},{"Start":"02:06.485 ","End":"02:08.363","Text":"Then we, again,"},{"Start":"02:08.363 ","End":"02:10.080","Text":"get the 4 unknowns,"},{"Start":"02:10.080 ","End":"02:11.250","Text":"but 2 of them are known."},{"Start":"02:11.250 ","End":"02:12.870","Text":"B is 1 and D is 1,"},{"Start":"02:12.870 ","End":"02:15.725","Text":"so that gives us another equation in A and C,"},{"Start":"02:15.725 ","End":"02:18.775","Text":"and we get this."},{"Start":"02:18.775 ","End":"02:22.710","Text":"We have now 2 equations and 2 unknowns."},{"Start":"02:22.710 ","End":"02:23.955","Text":"I take these 2,"},{"Start":"02:23.955 ","End":"02:27.780","Text":"put them as a system of 2 equations and 2 unknowns A and C,"},{"Start":"02:27.780 ","End":"02:29.900","Text":"and you know how to solve these,"},{"Start":"02:29.900 ","End":"02:33.630","Text":"but what you might want to do is to isolate one of them and say,"},{"Start":"02:33.630 ","End":"02:37.610","Text":"from here I took C equals 2 minus 2a by"},{"Start":"02:37.610 ","End":"02:41.900","Text":"bringing this over and then plugging C into here,"},{"Start":"02:41.900 ","End":"02:44.090","Text":"and that gives us this equation."},{"Start":"02:44.090 ","End":"02:45.750","Text":"If you tidy it up,"},{"Start":"02:45.750 ","End":"02:46.875","Text":"you get that A is 2."},{"Start":"02:46.875 ","End":"02:48.030","Text":"Once you have A is 2,"},{"Start":"02:48.030 ","End":"02:51.060","Text":"you put A=2 in here,"},{"Start":"02:51.060 ","End":"02:53.340","Text":"and that gives us C is 2 minus twice 2,"},{"Start":"02:53.340 ","End":"02:54.615","Text":"which is minus 2,"},{"Start":"02:54.615 ","End":"02:59.115","Text":"and so we have now A and C. We have B, we have D,"},{"Start":"02:59.115 ","End":"03:00.480","Text":"we have A,"},{"Start":"03:00.480 ","End":"03:02.070","Text":"and we have C. Now,"},{"Start":"03:02.070 ","End":"03:04.745","Text":"put all of those 4 constants in here."},{"Start":"03:04.745 ","End":"03:10.505","Text":"We get this expression and we recall that we want the Inverse Laplace Transform of it."},{"Start":"03:10.505 ","End":"03:13.100","Text":"Next, we use linearity."},{"Start":"03:13.100 ","End":"03:14.810","Text":"Just for reference, I\u0027m quoting"},{"Start":"03:14.810 ","End":"03:19.130","Text":"the linearity rule for 2 terms or it works for 4 terms also."},{"Start":"03:19.130 ","End":"03:23.360","Text":"Then we break this up and then we need 4 separate Laplace transforms."},{"Start":"03:23.360 ","End":"03:29.445","Text":"We go to our table of inverse transforms and see what looks closest to these."},{"Start":"03:29.445 ","End":"03:31.155","Text":"We find these 2 rules;"},{"Start":"03:31.155 ","End":"03:35.430","Text":"a rule for 1 over s minus a and the rule for 1 over s minus a^2."},{"Start":"03:35.430 ","End":"03:37.455","Text":"Obviously, you\u0027re going to use each one twice,"},{"Start":"03:37.455 ","End":"03:41.800","Text":"once with a=1 and once with a=2."},{"Start":"03:41.800 ","End":"03:48.990","Text":"In short, we end up with this expression and we are done."}],"ID":8022},{"Watched":false,"Name":"Exercise 21","Duration":"3m 40s","ChapterTopicVideoID":7945,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this exercise, we want to compute"},{"Start":"00:02.220 ","End":"00:07.320","Text":"the Inverse Laplace Transform of this 1 over s^2 plus 2x plus 3,"},{"Start":"00:07.320 ","End":"00:11.070","Text":"and at first, you might think it\u0027s a case for partial fractions,"},{"Start":"00:11.070 ","End":"00:14.505","Text":"but the thing is that the denominator doesn\u0027t factorize."},{"Start":"00:14.505 ","End":"00:17.130","Text":"If you tried to solve the quadratic equation, this equals 0."},{"Start":"00:17.130 ","End":"00:19.050","Text":"It has no roots, no real roots,"},{"Start":"00:19.050 ","End":"00:21.540","Text":"it has complex roots, but no real roots."},{"Start":"00:21.540 ","End":"00:23.790","Text":"What we do in a case like this is something called"},{"Start":"00:23.790 ","End":"00:26.730","Text":"completing the square technique from algebra,"},{"Start":"00:26.730 ","End":"00:29.310","Text":"which basically says what it says here,"},{"Start":"00:29.310 ","End":"00:31.830","Text":"but let\u0027s just see in our case."},{"Start":"00:31.830 ","End":"00:35.580","Text":"Here we have s^2 plus 2s, that\u0027s the beginning."},{"Start":"00:35.580 ","End":"00:39.615","Text":"I want to write this as something squared and then adjust it."},{"Start":"00:39.615 ","End":"00:42.330","Text":"Now if it\u0027s going to be s plus something squared,"},{"Start":"00:42.330 ","End":"00:44.420","Text":"it\u0027ll be s plus 1^2 because this has to be"},{"Start":"00:44.420 ","End":"00:49.490","Text":"twice the last term from the a^2 plus 2ab plus b^2, you know what I mean."},{"Start":"00:49.490 ","End":"00:52.550","Text":"It has to be plus 1^2, but if I square this,"},{"Start":"00:52.550 ","End":"00:54.880","Text":"I\u0027ve got s^2 plus 2s plus 1,"},{"Start":"00:54.880 ","End":"00:56.940","Text":"so that\u0027s too much,"},{"Start":"00:56.940 ","End":"00:59.840","Text":"so I need to compensate by subtracting 1."},{"Start":"00:59.840 ","End":"01:01.760","Text":"That\u0027s basically what this rule says."},{"Start":"01:01.760 ","End":"01:05.510","Text":"It\u0027s going to be s plus half whatever this is in case half of this"},{"Start":"01:05.510 ","End":"01:09.605","Text":"is 1 and then squared and then minus the compensation."},{"Start":"01:09.605 ","End":"01:11.590","Text":"Now if I put that back in here,"},{"Start":"01:11.590 ","End":"01:15.360","Text":"I get s plus 1^2 minus 1, then there\u0027s a plus 3,"},{"Start":"01:15.360 ","End":"01:16.940","Text":"and I can combine this,"},{"Start":"01:16.940 ","End":"01:21.100","Text":"and then I will get this expression."},{"Start":"01:21.100 ","End":"01:23.120","Text":"Now, this is still not good,"},{"Start":"01:23.120 ","End":"01:24.905","Text":"at least in my table,"},{"Start":"01:24.905 ","End":"01:27.590","Text":"I don\u0027t find anything that quite matches this."},{"Start":"01:27.590 ","End":"01:32.490","Text":"I do have something that\u0027s 1 over s^2 plus a^2."},{"Start":"01:32.490 ","End":"01:37.455","Text":"If I could just take this s plus 1 and somehow replace it by s,"},{"Start":"01:37.455 ","End":"01:39.180","Text":"that would be nice."},{"Start":"01:39.180 ","End":"01:42.445","Text":"Unfortunately, there is a rule that helps us to do that,"},{"Start":"01:42.445 ","End":"01:44.850","Text":"and this is the rule I\u0027m going to use."},{"Start":"01:44.850 ","End":"01:48.830","Text":"There\u0027s a typo, this should be big F, sorry about that."},{"Start":"01:48.830 ","End":"01:52.960","Text":"I need a big F and a big F. Now,"},{"Start":"01:52.960 ","End":"01:58.925","Text":"you\u0027re wondering maybe why there is an s minus 1 here when I was talking about s plus 1,"},{"Start":"01:58.925 ","End":"02:01.405","Text":"well, you\u0027ll see it actually works out fine,"},{"Start":"02:01.405 ","End":"02:06.870","Text":"because if I let this function here be big F(s),"},{"Start":"02:06.870 ","End":"02:12.840","Text":"then F of s minus 1 will be 1 over,"},{"Start":"02:12.840 ","End":"02:16.300","Text":"and then instead of s, I put s minus 1."},{"Start":"02:16.300 ","End":"02:22.335","Text":"I\u0027ll even put extra brackets plus 1^2 plus 2."},{"Start":"02:22.335 ","End":"02:25.875","Text":"Now you see that the minus 1 and the plus 1 cancel,"},{"Start":"02:25.875 ","End":"02:29.730","Text":"and this is just 1 over s^2 plus 2."},{"Start":"02:29.730 ","End":"02:32.445","Text":"This is the correct rule that we need,"},{"Start":"02:32.445 ","End":"02:37.100","Text":"the table of rules gave this a=1 in this rule."},{"Start":"02:37.100 ","End":"02:38.990","Text":"Now looking closer at this rule,"},{"Start":"02:38.990 ","End":"02:40.380","Text":"what it really says is,"},{"Start":"02:40.380 ","End":"02:45.165","Text":"\"You can replace s by s minus 1 in the function,"},{"Start":"02:45.165 ","End":"02:51.965","Text":"but you also have to compensate by putting this extra e to the minus t in front.\""},{"Start":"02:51.965 ","End":"02:55.985","Text":"We get this, here\u0027s the e to the minus t for compensation."},{"Start":"02:55.985 ","End":"02:58.785","Text":"It\u0027s e to the minus 1t because a is 1,"},{"Start":"02:58.785 ","End":"03:02.850","Text":"and here\u0027s the f of s minus 1 inside here."},{"Start":"03:02.850 ","End":"03:06.110","Text":"Now we have this and this is now very"},{"Start":"03:06.110 ","End":"03:10.145","Text":"good because we have something very close to this in the table."},{"Start":"03:10.145 ","End":"03:14.735","Text":"We have the Inverse Laplace Transform of 1 over s^2 plus a^2."},{"Start":"03:14.735 ","End":"03:16.550","Text":"This is not something squared,"},{"Start":"03:16.550 ","End":"03:23.050","Text":"but you can make it something squared by writing the 2 as a square root of 2^2."},{"Start":"03:23.050 ","End":"03:28.200","Text":"Now we can let a equals root 2 in this equation,"},{"Start":"03:28.200 ","End":"03:30.650","Text":"and that will give us e to the minus t,"},{"Start":"03:30.650 ","End":"03:34.100","Text":"1 over root t. The 1 over root t I brought out from"},{"Start":"03:34.100 ","End":"03:38.045","Text":"here to the front times sine root 2t,"},{"Start":"03:38.045 ","End":"03:40.620","Text":"and that\u0027s the answer."}],"ID":8023},{"Watched":false,"Name":"Exercise 22","Duration":"4m 50s","ChapterTopicVideoID":7946,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this exercise, we want to compute"},{"Start":"00:01.980 ","End":"00:06.780","Text":"the Inverse Laplace Transform of 1 over s^2 plus s plus 1."},{"Start":"00:06.780 ","End":"00:10.350","Text":"In my table I don\u0027t see anything quite like this."},{"Start":"00:10.350 ","End":"00:12.705","Text":"We\u0027re going to have to do a bit of algebra first."},{"Start":"00:12.705 ","End":"00:15.590","Text":"Now this denominator does not factorize."},{"Start":"00:15.590 ","End":"00:17.180","Text":"If you check for the roots,"},{"Start":"00:17.180 ","End":"00:18.890","Text":"it doesn\u0027t have any real roots."},{"Start":"00:18.890 ","End":"00:20.150","Text":"Yeah, it has complex roots,"},{"Start":"00:20.150 ","End":"00:25.820","Text":"but not real roots and so what we do is something called completing the square."},{"Start":"00:25.820 ","End":"00:28.400","Text":"There is a formula and I\u0027ll use the formula,"},{"Start":"00:28.400 ","End":"00:32.135","Text":"but later I\u0027ll show you how to go around the formula."},{"Start":"00:32.135 ","End":"00:37.330","Text":"In general, this is what I have and what I\u0027m going to do is I see s^2 plus bs,"},{"Start":"00:37.330 ","End":"00:38.710","Text":"I see s^2 plus s,"},{"Start":"00:38.710 ","End":"00:42.765","Text":"so I\u0027ll take b=1 and then apply this."},{"Start":"00:42.765 ","End":"00:45.540","Text":"I get s^2 plus s, which is 1s,"},{"Start":"00:45.540 ","End":"00:47.839","Text":"according to the formula,"},{"Start":"00:47.839 ","End":"00:51.945","Text":"s plus 1/2 squared minus 1/2 squared is 1/4."},{"Start":"00:51.945 ","End":"00:56.360","Text":"What we get, instead of this is in the denominator,"},{"Start":"00:56.360 ","End":"01:00.260","Text":"I can write s plus 1/2 squared minus 1/4 from here,"},{"Start":"01:00.260 ","End":"01:02.435","Text":"and then plus the 1 from here."},{"Start":"01:02.435 ","End":"01:05.615","Text":"Because minus 1/4 plus 1 is 3/4,"},{"Start":"01:05.615 ","End":"01:07.745","Text":"I can rewrite it like this."},{"Start":"01:07.745 ","End":"01:11.705","Text":"Before I continue, I wanted to show you how we could get without the formula."},{"Start":"01:11.705 ","End":"01:13.925","Text":"We had s^2 plus s plus 1,"},{"Start":"01:13.925 ","End":"01:19.025","Text":"so what we do is we see this middle coefficient."},{"Start":"01:19.025 ","End":"01:21.445","Text":"It\u0027s not written here, but it\u0027s a 1."},{"Start":"01:21.445 ","End":"01:26.325","Text":"We have a 1 here and we take 1/2 of that, which is 1/2,"},{"Start":"01:26.325 ","End":"01:31.490","Text":"and write s plus 1/2 and then put it in brackets and write its squared."},{"Start":"01:31.490 ","End":"01:32.690","Text":"Now if we square this,"},{"Start":"01:32.690 ","End":"01:41.554","Text":"we get s^2 plus s plus 1/4 using the formula of a plus b squared binomial expansion."},{"Start":"01:41.554 ","End":"01:43.955","Text":"Now I look at this and I look at this."},{"Start":"01:43.955 ","End":"01:46.110","Text":"Assume here I have 1,"},{"Start":"01:46.110 ","End":"01:47.445","Text":"here I have 1/4."},{"Start":"01:47.445 ","End":"01:51.060","Text":"If I add 3/4 here, I would get this,"},{"Start":"01:51.060 ","End":"01:57.560","Text":"so I can write that this is s plus 1/2 squared and then I add the missing bit from"},{"Start":"01:57.560 ","End":"02:04.190","Text":"here to here plus 3/4 and that will get me to there right away without the formula."},{"Start":"02:04.190 ","End":"02:05.555","Text":"Just have to remember to take 1/2"},{"Start":"02:05.555 ","End":"02:09.935","Text":"this middle coefficient and then square it and then complete the difference."},{"Start":"02:09.935 ","End":"02:12.185","Text":"Back here and continuing,"},{"Start":"02:12.185 ","End":"02:15.050","Text":"this is still not something I have in my table at any rate."},{"Start":"02:15.050 ","End":"02:19.190","Text":"But if I had just s here instead of s plus 1/2, in other words,"},{"Start":"02:19.190 ","End":"02:24.650","Text":"if I could find some way to go from s plus 1/2 to just s,"},{"Start":"02:24.650 ","End":"02:28.055","Text":"then I could use the formula for 1 over s^2 plus"},{"Start":"02:28.055 ","End":"02:32.930","Text":"a^2 which is in the table and I could make this to be something squared."},{"Start":"02:32.930 ","End":"02:36.875","Text":"That\u0027s my goal to get this to be just s^2."},{"Start":"02:36.875 ","End":"02:40.190","Text":"Now there\u0027s a rule that will help us to do what we want to do."},{"Start":"02:40.190 ","End":"02:43.010","Text":"This is the rule I\u0027ll explain in a moment."},{"Start":"02:43.010 ","End":"02:47.045","Text":"But notice that it\u0027s in terms of a parameter a."},{"Start":"02:47.045 ","End":"02:51.815","Text":"In our case, we\u0027ll use it with a equaling 1/2."},{"Start":"02:51.815 ","End":"02:54.860","Text":"You might wonder, there\u0027s a minus 1/2,"},{"Start":"02:54.860 ","End":"02:56.870","Text":"but we have here a plus 1/2."},{"Start":"02:56.870 ","End":"03:02.165","Text":"Well, that\u0027s exactly what makes it work out and we\u0027ll see this in just a moment."},{"Start":"03:02.165 ","End":"03:05.870","Text":"But I just want to interpret what this rule says."},{"Start":"03:05.870 ","End":"03:08.120","Text":"Let\u0027s say in our case with the 1/2,"},{"Start":"03:08.120 ","End":"03:13.730","Text":"it says that if I have to compute the inverse transform of a function s, F,"},{"Start":"03:13.730 ","End":"03:18.545","Text":"it\u0027s okay if I replace s by s minus 1/2,"},{"Start":"03:18.545 ","End":"03:23.930","Text":"as long as I compensate by putting e to the minus 1/2t in front."},{"Start":"03:23.930 ","End":"03:25.295","Text":"That\u0027s one way of looking at it."},{"Start":"03:25.295 ","End":"03:28.355","Text":"This is the adjustment and this is the compensation."},{"Start":"03:28.355 ","End":"03:34.020","Text":"If I do that here and we replace s by s minus 1/2,"},{"Start":"03:34.020 ","End":"03:36.035","Text":"then we will get what we want."},{"Start":"03:36.035 ","End":"03:39.845","Text":"Because if I replace s by s minus 1/2 here,"},{"Start":"03:39.845 ","End":"03:47.140","Text":"I\u0027ll get s minus 1/2 plus 1/2 squared,"},{"Start":"03:47.140 ","End":"03:49.355","Text":"and s minus 1/2 plus 1/2 is just s^2."},{"Start":"03:49.355 ","End":"03:51.410","Text":"That\u0027s why I took the opposite sign."},{"Start":"03:51.410 ","End":"03:56.440","Text":"I need the minus 1/2 because I have a plus 1/2 and then it cancels out."},{"Start":"03:56.440 ","End":"04:01.290","Text":"Now we\u0027re at the point of 1 over s^2 plus 3/4."},{"Start":"04:01.290 ","End":"04:05.115","Text":"Remember I said there\u0027s a rule for 1 over s^2 plus a^2."},{"Start":"04:05.115 ","End":"04:08.870","Text":"What I can do is rewrite the 3/4 as something squared."},{"Start":"04:08.870 ","End":"04:11.240","Text":"But in a moment, I just want to let you know what"},{"Start":"04:11.240 ","End":"04:16.485","Text":"the inverse transform of this is in general and this is the rule."},{"Start":"04:16.485 ","End":"04:19.560","Text":"Now, we have instead of a^2, we have 3/4,"},{"Start":"04:19.560 ","End":"04:26.585","Text":"so if a^2 is 3/4 then a is just the square root of 3/4."},{"Start":"04:26.585 ","End":"04:30.090","Text":"Then I can use this general rule,"},{"Start":"04:30.090 ","End":"04:34.415","Text":"and if I substitute this a equals root 3/4,"},{"Start":"04:34.415 ","End":"04:36.520","Text":"I will get 1/a,"},{"Start":"04:36.520 ","End":"04:38.910","Text":"which is 1 over this root of 3/4."},{"Start":"04:38.910 ","End":"04:40.370","Text":"It normally be here,"},{"Start":"04:40.370 ","End":"04:41.960","Text":"but I brought it to the front,"},{"Start":"04:41.960 ","End":"04:45.740","Text":"then the e to the minus 1/2t and here the sine at,"},{"Start":"04:45.740 ","End":"04:48.230","Text":"which is sine root 3/4t,"},{"Start":"04:48.230 ","End":"04:50.730","Text":"and we are done."}],"ID":8024},{"Watched":false,"Name":"Exercise 23","Duration":"2m 45s","ChapterTopicVideoID":7947,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.635","Text":"In this exercise, we want to compute the Inverse Laplace Transform of this expression."},{"Start":"00:05.635 ","End":"00:07.090","Text":"It\u0027s a rational expression."},{"Start":"00:07.090 ","End":"00:09.025","Text":"We have a cubic on the denominator,"},{"Start":"00:09.025 ","End":"00:11.335","Text":"a quadratic on the numerator."},{"Start":"00:11.335 ","End":"00:14.650","Text":"As usual, we\u0027ll be doing this with partial fractions."},{"Start":"00:14.650 ","End":"00:19.885","Text":"Note that the denominator is already decomposed and factored it as much as we can go."},{"Start":"00:19.885 ","End":"00:21.640","Text":"s^2 plus 1 can\u0027t be factored."},{"Start":"00:21.640 ","End":"00:22.840","Text":"It has no roots,"},{"Start":"00:22.840 ","End":"00:24.295","Text":"at least within real numbers."},{"Start":"00:24.295 ","End":"00:29.785","Text":"So the general shape of the partial fraction is this."},{"Start":"00:29.785 ","End":"00:30.910","Text":"For the linear term,"},{"Start":"00:30.910 ","End":"00:36.265","Text":"we have a constant and for the irreducible quadratic we have a linear As plus B."},{"Start":"00:36.265 ","End":"00:37.630","Text":"So we\u0027ve got 3 constants,"},{"Start":"00:37.630 ","End":"00:39.265","Text":"A, B, and C to find."},{"Start":"00:39.265 ","End":"00:42.620","Text":"As usual, we multiply by the denominator on the left."},{"Start":"00:42.620 ","End":"00:45.275","Text":"This gives us this equation,"},{"Start":"00:45.275 ","End":"00:47.570","Text":"which is not really an equation,"},{"Start":"00:47.570 ","End":"00:51.110","Text":"it\u0027s an identity in S and we have to find what A,"},{"Start":"00:51.110 ","End":"00:52.610","Text":"B, and C are."},{"Start":"00:52.610 ","End":"00:54.950","Text":"Substitution is our main technique."},{"Start":"00:54.950 ","End":"00:56.810","Text":"If we put s=3,"},{"Start":"00:56.810 ","End":"01:01.110","Text":"that will be nice because that will make this 0 and we\u0027ll straight"},{"Start":"01:01.110 ","End":"01:06.370","Text":"away be able to get an equation in just C and get that C is 2."},{"Start":"01:06.370 ","End":"01:09.480","Text":"As for A and B, we substitute any two other values."},{"Start":"01:09.480 ","End":"01:12.935","Text":"s=0 because it will be easier to compute."},{"Start":"01:12.935 ","End":"01:16.640","Text":"That will give us on the left minus 1, on the right,"},{"Start":"01:16.640 ","End":"01:22.035","Text":"we\u0027ll get minus 3B plus C. We already know that C is 2,"},{"Start":"01:22.035 ","End":"01:24.495","Text":"so that gives us that B=1."},{"Start":"01:24.495 ","End":"01:26.190","Text":"Now we try another value,"},{"Start":"01:26.190 ","End":"01:30.575","Text":"s=1 should be pretty simple and this will give us this."},{"Start":"01:30.575 ","End":"01:34.835","Text":"But of course, we know that C is 2 and B is 1."},{"Start":"01:34.835 ","End":"01:36.140","Text":"So this will give us A,"},{"Start":"01:36.140 ","End":"01:38.165","Text":"which comes out to be 0."},{"Start":"01:38.165 ","End":"01:39.590","Text":"Then I take C, B,"},{"Start":"01:39.590 ","End":"01:43.755","Text":"and A and plug them into here. This is what we get."},{"Start":"01:43.755 ","End":"01:47.750","Text":"We remember that we\u0027re trying to find the Inverse Laplace Transform of this thing."},{"Start":"01:47.750 ","End":"01:51.640","Text":"So we\u0027ve now got to look up in the table."},{"Start":"01:51.640 ","End":"01:55.400","Text":"Let me say this, first let\u0027s decompose it according to linearity."},{"Start":"01:55.400 ","End":"01:58.010","Text":"For reference, this is the linearity property,"},{"Start":"01:58.010 ","End":"02:00.545","Text":"but we know how to work it in practice."},{"Start":"02:00.545 ","End":"02:03.230","Text":"Let me see if we can break it up into two separate pieces"},{"Start":"02:03.230 ","End":"02:06.135","Text":"and put L minus 1 on each of them."},{"Start":"02:06.135 ","End":"02:08.615","Text":"Also we can take the 2 as a constant in front."},{"Start":"02:08.615 ","End":"02:10.070","Text":"So we end up with this."},{"Start":"02:10.070 ","End":"02:16.100","Text":"Now these are two pretty basic things that we do find in the table or just about,"},{"Start":"02:16.100 ","End":"02:17.930","Text":"I mean, we don\u0027t have them exactly,"},{"Start":"02:17.930 ","End":"02:20.225","Text":"but we have in terms of a parameter a,"},{"Start":"02:20.225 ","End":"02:23.165","Text":"this one fits the paradigm of 1 over s minus a."},{"Start":"02:23.165 ","End":"02:28.700","Text":"If we let a=3 in this and here also,"},{"Start":"02:28.700 ","End":"02:29.960","Text":"if we let a=1,"},{"Start":"02:29.960 ","End":"02:33.080","Text":"then we\u0027ve got what we want. This is what we get."},{"Start":"02:33.080 ","End":"02:38.795","Text":"We get this sine at with a=1 and the 2 from here."},{"Start":"02:38.795 ","End":"02:43.400","Text":"Then we have the e^at with a=3, which is this."},{"Start":"02:43.400 ","End":"02:46.050","Text":"This is the answer and we\u0027re done."}],"ID":8025},{"Watched":false,"Name":"Exercise 24","Duration":"2m 48s","ChapterTopicVideoID":7953,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.610","Text":"Here, we have to find the Inverse Laplace Transform of this rational expression."},{"Start":"00:05.610 ","End":"00:08.880","Text":"We\u0027re going to do it with partial fractions as usual."},{"Start":"00:08.880 ","End":"00:11.700","Text":"Note that the denominator is fully factored."},{"Start":"00:11.700 ","End":"00:13.350","Text":"There\u0027s no more decomposition."},{"Start":"00:13.350 ","End":"00:15.720","Text":"s^2 plus 1 can\u0027t be factored,"},{"Start":"00:15.720 ","End":"00:17.670","Text":"at least not in terms of real numbers."},{"Start":"00:17.670 ","End":"00:22.020","Text":"These are the basic building blocks and we can use partial fractions."},{"Start":"00:22.020 ","End":"00:23.940","Text":"Then the general shape will be this."},{"Start":"00:23.940 ","End":"00:25.470","Text":"For the irreducible quadratic,"},{"Start":"00:25.470 ","End":"00:27.810","Text":"we have a linear term and for the linear,"},{"Start":"00:27.810 ","End":"00:29.400","Text":"we get a constant term."},{"Start":"00:29.400 ","End":"00:32.790","Text":"We need to find A, B, and C. I guess I should have mentioned,"},{"Start":"00:32.790 ","End":"00:35.315","Text":"though it\u0027s fairly clear that for partial fractions,"},{"Start":"00:35.315 ","End":"00:39.020","Text":"we have to have a degree in the numerator lower than the denominator."},{"Start":"00:39.020 ","End":"00:42.245","Text":"This is degree 2, this is degree 3, so we\u0027re okay."},{"Start":"00:42.245 ","End":"00:43.850","Text":"Now our usual technique,"},{"Start":"00:43.850 ","End":"00:47.090","Text":"we multiply out by this and we get this,"},{"Start":"00:47.090 ","End":"00:49.580","Text":"which is an identity in s. It\u0027s going to be"},{"Start":"00:49.580 ","End":"00:52.300","Text":"true for all s. We want to find the constants A,"},{"Start":"00:52.300 ","End":"00:56.290","Text":"B, and C. Now we can substitute any value of s we want."},{"Start":"00:56.290 ","End":"00:59.540","Text":"We often choose something that\u0027ll make a factor disappear."},{"Start":"00:59.540 ","End":"01:01.980","Text":"Like if I plug in minus 2,"},{"Start":"01:01.980 ","End":"01:06.585","Text":"then this term disappears and this whole thing disappears. 0 times something."},{"Start":"01:06.585 ","End":"01:08.025","Text":"s^2 plus 1 is 5."},{"Start":"01:08.025 ","End":"01:13.720","Text":"We get this equation and we find C. Now we just plug in any two other values."},{"Start":"01:13.720 ","End":"01:16.070","Text":"I often choose 0. It\u0027s convenient."},{"Start":"01:16.070 ","End":"01:20.780","Text":"Plug 0 in and we get 1 equals 2B plus C. But remember that we already found C,"},{"Start":"01:20.780 ","End":"01:25.030","Text":"which is 1, and that gives us that B is 0."},{"Start":"01:25.030 ","End":"01:27.170","Text":"Finally, substitute another value,"},{"Start":"01:27.170 ","End":"01:29.929","Text":"or like s equals 1, if it\u0027s available."},{"Start":"01:29.929 ","End":"01:31.565","Text":"That\u0027s easy to work with."},{"Start":"01:31.565 ","End":"01:34.880","Text":"Though, we get this equation and it\u0027s got A, B, and C in it,"},{"Start":"01:34.880 ","End":"01:36.425","Text":"but we\u0027ve already found Cc,"},{"Start":"01:36.425 ","End":"01:40.230","Text":"which is 1, and we\u0027ve already found B, which is 0."},{"Start":"01:40.230 ","End":"01:43.425","Text":"You do the math, you get A is 1 here."},{"Start":"01:43.425 ","End":"01:47.385","Text":"Now we have A, B, and C. We plug them in here."},{"Start":"01:47.385 ","End":"01:50.690","Text":"Now we can rewrite this in more decomposed terms,"},{"Start":"01:50.690 ","End":"01:52.340","Text":"in partial fractions,"},{"Start":"01:52.340 ","End":"01:56.370","Text":"and what we get is, A is 1,"},{"Start":"01:56.370 ","End":"02:00.735","Text":"B is 0, so As plus B is just s and C is 1."},{"Start":"02:00.735 ","End":"02:02.960","Text":"This is the expression we get."},{"Start":"02:02.960 ","End":"02:06.260","Text":"Now we turn to our table of"},{"Start":"02:06.260 ","End":"02:11.570","Text":"Inverse Laplace Transforms and look for whatever is closest in a moment,"},{"Start":"02:11.570 ","End":"02:13.370","Text":"I forgot to say that we\u0027re going to, of course,"},{"Start":"02:13.370 ","End":"02:16.730","Text":"use linearity of the transform to break it up."},{"Start":"02:16.730 ","End":"02:19.520","Text":"This is the abstract rule for linearity, but we know what it means."},{"Start":"02:19.520 ","End":"02:21.215","Text":"We can just break it up into pieces."},{"Start":"02:21.215 ","End":"02:23.690","Text":"Now we have two inverse transforms."},{"Start":"02:23.690 ","End":"02:25.655","Text":"This one and this one,"},{"Start":"02:25.655 ","End":"02:27.445","Text":"we have in the table."},{"Start":"02:27.445 ","End":"02:28.950","Text":"Well, at least something very similar."},{"Start":"02:28.950 ","End":"02:34.950","Text":"Like this is the same as this with a=1 and this is the same as this with a=2."},{"Start":"02:34.950 ","End":"02:41.670","Text":"If we take cosine at and put a=1 and e to the minus at with a=2 and plug them in here,"},{"Start":"02:41.670 ","End":"02:49.180","Text":"what we get is this cosine 1t plus e to the minus 2t and, that\u0027s the answer."}],"ID":8026},{"Watched":false,"Name":"Exercise 25","Duration":"4m 11s","ChapterTopicVideoID":7954,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.455","Text":"Here, we have another Inverse Laplace Transform to calculate."},{"Start":"00:04.455 ","End":"00:10.545","Text":"This time, we have this expression and we\u0027re going to deal with partial fractions."},{"Start":"00:10.545 ","End":"00:12.510","Text":"If you look at the denominator,"},{"Start":"00:12.510 ","End":"00:16.170","Text":"you\u0027ll see that it\u0027s already fully factored because s^2"},{"Start":"00:16.170 ","End":"00:20.835","Text":"plus 1 has no real roots and s^2 plus 4 has no real roots,"},{"Start":"00:20.835 ","End":"00:22.980","Text":"they\u0027re both irreducible and so"},{"Start":"00:22.980 ","End":"00:26.790","Text":"the general form of the partial fraction will be like this."},{"Start":"00:26.790 ","End":"00:30.360","Text":"We need the linear term for each of the 2 irreducible quadratics,"},{"Start":"00:30.360 ","End":"00:31.800","Text":"As plus B here,"},{"Start":"00:31.800 ","End":"00:35.580","Text":"Cs plus D here and our task is now to find the constants A,"},{"Start":"00:35.580 ","End":"00:39.255","Text":"B, C, and D that will make this into an identity."},{"Start":"00:39.255 ","End":"00:43.160","Text":"As usual we get rid of of all the denominators if"},{"Start":"00:43.160 ","End":"00:46.835","Text":"we multiply by this and this is what we get."},{"Start":"00:46.835 ","End":"00:49.550","Text":"Now, of course, we could just substitute"},{"Start":"00:49.550 ","End":"00:53.480","Text":"4 different values and then get 4 equations and 4 unknowns,"},{"Start":"00:53.480 ","End":"00:57.680","Text":"but hopefully we might be able to simplify a bit to make the work easier."},{"Start":"00:57.680 ","End":"01:01.240","Text":"There is no value that we can substitute to make s^2 plus 1,"},{"Start":"01:01.240 ","End":"01:04.430","Text":"0, x^2 plus 4, 0, they\u0027re both irreducible."},{"Start":"01:04.430 ","End":"01:06.590","Text":"Let me show you what we might do."},{"Start":"01:06.590 ","End":"01:09.575","Text":"Multiply everything else and open the brackets."},{"Start":"01:09.575 ","End":"01:16.850","Text":"Next we can collect like terms."},{"Start":"01:16.850 ","End":"01:19.445","Text":"As far as powers of s go constants,"},{"Start":"01:19.445 ","End":"01:20.720","Text":"terms containing s, s^2,"},{"Start":"01:20.720 ","End":"01:24.125","Text":"and s^3, we got this."},{"Start":"01:24.125 ","End":"01:26.884","Text":"Now if 2 polynomials are equal,"},{"Start":"01:26.884 ","End":"01:29.855","Text":"then all their coefficients are equal,"},{"Start":"01:29.855 ","End":"01:32.675","Text":"that will give us already 4 equations."},{"Start":"01:32.675 ","End":"01:34.900","Text":"Make some more space here."},{"Start":"01:34.900 ","End":"01:36.620","Text":"If we compare the constants,"},{"Start":"01:36.620 ","End":"01:42.005","Text":"we get 3 is 4B plus D. All the other powers are going to be 0,"},{"Start":"01:42.005 ","End":"01:47.960","Text":"the s^1, s^2, s^3, we\u0027ve got 0=0=0 equals this,"},{"Start":"01:47.960 ","End":"01:49.070","Text":"then this, then this."},{"Start":"01:49.070 ","End":"01:51.740","Text":"Maybe you have 4 equations and 4 unknowns."},{"Start":"01:51.740 ","End":"01:53.465","Text":"But if you notice in most of them,"},{"Start":"01:53.465 ","End":"01:55.130","Text":"there\u0027s only 2 unknowns,"},{"Start":"01:55.130 ","End":"01:56.800","Text":"so it\u0027ll make things easier."},{"Start":"01:56.800 ","End":"02:00.410","Text":"This is not too hard to solve and present the answer."},{"Start":"02:00.410 ","End":"02:02.735","Text":"Let me show you maybe how I got B."},{"Start":"02:02.735 ","End":"02:07.755","Text":"If you take this equation and subtract this equation,"},{"Start":"02:07.755 ","End":"02:10.275","Text":"then we\u0027ll get 3=3B,"},{"Start":"02:10.275 ","End":"02:11.790","Text":"so B is 1."},{"Start":"02:11.790 ","End":"02:13.110","Text":"Plus B=1 here,"},{"Start":"02:13.110 ","End":"02:15.300","Text":"you get D equals minus 1."},{"Start":"02:15.300 ","End":"02:17.505","Text":"Then similarly here, if you subtract,"},{"Start":"02:17.505 ","End":"02:19.010","Text":"we\u0027ve got 3A is 0,"},{"Start":"02:19.010 ","End":"02:21.370","Text":"so A is 0 and then C is 0."},{"Start":"02:21.370 ","End":"02:30.620","Text":"I guess I showed you how we did it and now we\u0027re going to plug in A,"},{"Start":"02:30.620 ","End":"02:32.660","Text":"B, C, and D that we found."},{"Start":"02:32.660 ","End":"02:38.405","Text":"Well the A and C is 0 and we\u0027ll just get the constants here and show it."},{"Start":"02:38.405 ","End":"02:41.180","Text":"We get the 1 and the minus 1 and we remember"},{"Start":"02:41.180 ","End":"02:44.675","Text":"that we were looking for the inverse transform of this thing."},{"Start":"02:44.675 ","End":"02:48.780","Text":"Now it\u0027s time to go to the lookup table because each"},{"Start":"02:48.780 ","End":"02:52.955","Text":"of these can be found there or at least close to it."},{"Start":"02:52.955 ","End":"02:56.765","Text":"But, of course, first we use linearity to split it up into 2 bits."},{"Start":"02:56.765 ","End":"02:58.760","Text":"For reference, this is the linearity rule."},{"Start":"02:58.760 ","End":"03:00.095","Text":"We don\u0027t really need it."},{"Start":"03:00.095 ","End":"03:03.500","Text":"This breaks up into this plus this but the minus comes out in front,"},{"Start":"03:03.500 ","End":"03:05.420","Text":"so we\u0027ve got this minus this."},{"Start":"03:05.420 ","End":"03:10.730","Text":"Now these 2 can be found in the table using the following rule where"},{"Start":"03:10.730 ","End":"03:17.020","Text":"here we\u0027re going to let A=1 but here we\u0027re gonna let A=2."},{"Start":"03:17.020 ","End":"03:22.970","Text":"Some tables have seen a slight variation of this where they say L to the minus 1 of 1"},{"Start":"03:22.970 ","End":"03:29.400","Text":"over s^2 plus A^2 is 1 over A sine at."},{"Start":"03:29.400 ","End":"03:34.005","Text":"It\u0027s essentially the same but the A was rounded to the denominator."},{"Start":"03:34.005 ","End":"03:37.370","Text":"Actually this would have been easier but let\u0027s work with this one which gets"},{"Start":"03:37.370 ","End":"03:40.760","Text":"have to tweak this a little bit then need to put a 2 here."},{"Start":"03:40.760 ","End":"03:44.420","Text":"4 is 2^2 so we rewrite this like this,"},{"Start":"03:44.420 ","End":"03:49.040","Text":"put here a 2 and write the 1 as 1^2 or write 4 as 2^2."},{"Start":"03:49.040 ","End":"03:51.880","Text":"Also because I put a 2 here,"},{"Start":"03:51.880 ","End":"03:54.485","Text":"I compensate with a 1/2 here."},{"Start":"03:54.485 ","End":"04:00.290","Text":"This cancels with this so we haven\u0027t really done anything and this gives us the answer."},{"Start":"04:00.290 ","End":"04:04.715","Text":"We could have gotten to it a bit faster if we\u0027d use this formula. It all depends."},{"Start":"04:04.715 ","End":"04:09.850","Text":"Table slightly vary as to what they contain of inverse transforms, that is."},{"Start":"04:09.850 ","End":"04:12.060","Text":"Anyway this is the answer."}],"ID":8027},{"Watched":false,"Name":"Exercise 26","Duration":"5m 4s","ChapterTopicVideoID":7955,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this exercise, we\u0027re going to compute the inverse"},{"Start":"00:02.730 ","End":"00:05.730","Text":"Laplace transform of this expression,"},{"Start":"00:05.730 ","End":"00:09.420","Text":"1 over s times s^2 plus 1^2."},{"Start":"00:09.420 ","End":"00:10.920","Text":"As you probably guessed we\u0027ll be using"},{"Start":"00:10.920 ","End":"00:14.745","Text":"partial fractions to decompose this rational expression."},{"Start":"00:14.745 ","End":"00:19.350","Text":"Notice that the denominator is already fully factorized,"},{"Start":"00:19.350 ","End":"00:21.930","Text":"there\u0027s nothing more we can do to break it up."},{"Start":"00:21.930 ","End":"00:26.355","Text":"s^2 plus 1 doesn\u0027t have any roots, can\u0027t be factored."},{"Start":"00:26.355 ","End":"00:31.440","Text":"The general shape of the partial fraction is this."},{"Start":"00:31.440 ","End":"00:34.095","Text":"For the s we need a constant over s,"},{"Start":"00:34.095 ","End":"00:38.460","Text":"and s^2 plus 1^2 we need to take this thing"},{"Start":"00:38.460 ","End":"00:43.055","Text":"squared n^1 and the linear term on the top for each."},{"Start":"00:43.055 ","End":"00:44.240","Text":"If you\u0027re not sure about this,"},{"Start":"00:44.240 ","End":"00:47.780","Text":"go and review the theory on partial fractions."},{"Start":"00:47.780 ","End":"00:53.745","Text":"We got rid of the denominators by multiplying by this denominator on both sides."},{"Start":"00:53.745 ","End":"00:56.985","Text":"Our aim, of course, is to find the constants A, B, C, D,"},{"Start":"00:56.985 ","End":"01:01.785","Text":"E. Let\u0027s just simplify this a bit."},{"Start":"01:01.785 ","End":"01:05.460","Text":"Here I square the s^2 plus 1,"},{"Start":"01:05.460 ","End":"01:11.365","Text":"the s I throw inside this bracket and similarly here."},{"Start":"01:11.365 ","End":"01:16.775","Text":"By the way, the first step could\u0027ve been to let s=0."},{"Start":"01:16.775 ","End":"01:19.100","Text":"If we did that, this would have come out 0,"},{"Start":"01:19.100 ","End":"01:22.930","Text":"this would have come out 0 when we would\u0027ve got A=1."},{"Start":"01:22.930 ","End":"01:26.280","Text":"But we\u0027ll see this anyway the other way."},{"Start":"01:26.280 ","End":"01:28.845","Text":"There\u0027s more than 1 way to solve this."},{"Start":"01:28.845 ","End":"01:32.460","Text":"Now we continue expanding and"},{"Start":"01:32.460 ","End":"01:36.740","Text":"now all the brackets have opened and now we want to start collecting like terms."},{"Start":"01:36.740 ","End":"01:39.575","Text":"According to increasing powers of s,"},{"Start":"01:39.575 ","End":"01:41.600","Text":"constants just s^2, s^3,"},{"Start":"01:41.600 ","End":"01:46.430","Text":"s^4 what we\u0027re going to do is compare"},{"Start":"01:46.430 ","End":"01:51.695","Text":"powers of s. I have 2 polynomials on each side of the equals,"},{"Start":"01:51.695 ","End":"02:00.590","Text":"and I need to compare 2 polynomials the same then all the coefficients are the same."},{"Start":"02:00.590 ","End":"02:02.120","Text":"The left-hand side is 1,"},{"Start":"02:02.120 ","End":"02:03.890","Text":"so the right-hand side has to be 1."},{"Start":"02:03.890 ","End":"02:07.055","Text":"So all these coefficients are going to be 0,"},{"Start":"02:07.055 ","End":"02:09.040","Text":"but this one is going to be 1."},{"Start":"02:09.040 ","End":"02:12.250","Text":"Like we said before, A comes out to be 1."},{"Start":"02:12.250 ","End":"02:15.680","Text":"Anyway, we also have C plus E is 0 and so on,"},{"Start":"02:15.680 ","End":"02:20.520","Text":"all these coefficients are 0 and we\u0027ve also found C already, it\u0027s 0."},{"Start":"02:21.200 ","End":"02:23.580","Text":"From here, we can solve all of them."},{"Start":"02:23.580 ","End":"02:25.660","Text":"For example, or let\u0027s just do it all."},{"Start":"02:25.660 ","End":"02:27.760","Text":"A is 1, plug it in here,"},{"Start":"02:27.760 ","End":"02:30.035","Text":"we\u0027ve got B is minus 1."},{"Start":"02:30.035 ","End":"02:33.585","Text":"Then we already said that A is 1, C is 0."},{"Start":"02:33.585 ","End":"02:36.940","Text":"E we can get from plugging C=0 here,"},{"Start":"02:36.940 ","End":"02:42.930","Text":"so that\u0027s also 0 and D we can get by plugging A and B, both in here."},{"Start":"02:42.930 ","End":"02:45.120","Text":"A is 1, B is minus 1,"},{"Start":"02:45.120 ","End":"02:47.055","Text":"so we get D is minus 1."},{"Start":"02:47.055 ","End":"02:48.785","Text":"Now we have all the constants."},{"Start":"02:48.785 ","End":"02:53.815","Text":"So let\u0027s put them into the expression for the partial fractions."},{"Start":"02:53.815 ","End":"02:55.240","Text":"We get this, this,"},{"Start":"02:55.240 ","End":"02:56.560","Text":"and this, and of course,"},{"Start":"02:56.560 ","End":"02:59.695","Text":"we are still looking for the inverse transform of this."},{"Start":"02:59.695 ","End":"03:05.455","Text":"This is the partial fraction decomposition of the original expression."},{"Start":"03:05.455 ","End":"03:07.625","Text":"This is easier to work with."},{"Start":"03:07.625 ","End":"03:13.610","Text":"Now we go in and break it up further using linearity."},{"Start":"03:13.610 ","End":"03:16.190","Text":"Just break it up into pieces."},{"Start":"03:16.190 ","End":"03:22.955","Text":"For reference, I\u0027ve quoted the rule for linearity of the inverse transform."},{"Start":"03:22.955 ","End":"03:25.730","Text":"Here it\u0027s written for 2 terms,"},{"Start":"03:25.730 ","End":"03:29.705","Text":"but it works for 3 terms or so as in a similar way."},{"Start":"03:29.705 ","End":"03:32.150","Text":"It means that we just break each piece up and we can take"},{"Start":"03:32.150 ","End":"03:35.105","Text":"constants like minus 1 outside,"},{"Start":"03:35.105 ","End":"03:37.385","Text":"and we end up in this."},{"Start":"03:37.385 ","End":"03:42.410","Text":"At this point it\u0027s time to go and see what we have in our table"},{"Start":"03:42.410 ","End":"03:47.320","Text":"of inverse Laplace transforms and we find 3 rules that are going to help us."},{"Start":"03:47.320 ","End":"03:49.850","Text":"This one is this exactly."},{"Start":"03:49.850 ","End":"03:51.590","Text":"This one is this,"},{"Start":"03:51.590 ","End":"03:56.120","Text":"but we have to put A=1 here."},{"Start":"03:56.120 ","End":"03:59.260","Text":"Here it looks like this,"},{"Start":"03:59.260 ","End":"04:04.415","Text":"but we are going to have to make a little bit of adjustments,"},{"Start":"04:04.415 ","End":"04:08.305","Text":"because if I let A=1,"},{"Start":"04:08.305 ","End":"04:12.000","Text":"I don\u0027t get exactly this."},{"Start":"04:12.000 ","End":"04:15.880","Text":"I\u0027m going to get 2s instead of just s,"},{"Start":"04:15.880 ","End":"04:19.715","Text":"but we can easily make some adjustments."},{"Start":"04:19.715 ","End":"04:23.090","Text":"Anyway, the first term comes out to be just 1,"},{"Start":"04:23.090 ","End":"04:28.255","Text":"and the second term becomes minus"},{"Start":"04:28.255 ","End":"04:34.200","Text":"cosine t. We have cosine at with a=1 here,"},{"Start":"04:34.200 ","End":"04:35.520","Text":"so it\u0027s cosine 1t,"},{"Start":"04:35.520 ","End":"04:40.880","Text":"which is just cosine t. The last term in order to get it to be in this shape,"},{"Start":"04:40.880 ","End":"04:45.275","Text":"2as, a is 1 so I just need to put a 2 here."},{"Start":"04:45.275 ","End":"04:47.450","Text":"But if I put a 2 here, I need to compensate,"},{"Start":"04:47.450 ","End":"04:48.935","Text":"so I put 1/2 here."},{"Start":"04:48.935 ","End":"04:55.455","Text":"Now I do have that this is equal to this with a=1 and so we can finally"},{"Start":"04:55.455 ","End":"04:59.660","Text":"get this t sine at is"},{"Start":"04:59.660 ","End":"05:05.160","Text":"this and the 1/2 from here and this is the final answer, so we\u0027re done."}],"ID":8028},{"Watched":false,"Name":"Exercise 27","Duration":"4m 31s","ChapterTopicVideoID":7956,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this exercise, we\u0027re going to compute"},{"Start":"00:01.980 ","End":"00:05.430","Text":"the Inverse Laplace Transform of this expression,"},{"Start":"00:05.430 ","End":"00:08.025","Text":"1 over s times s^2 plus 1^2."},{"Start":"00:08.025 ","End":"00:09.480","Text":"As you probably guessed,"},{"Start":"00:09.480 ","End":"00:13.695","Text":"would be using partial fractions to decompose this rational expression."},{"Start":"00:13.695 ","End":"00:18.050","Text":"Notice that the denominator is already fully factorized,"},{"Start":"00:18.050 ","End":"00:20.645","Text":"there\u0027s nothing more we can do to break it up,"},{"Start":"00:20.645 ","End":"00:24.740","Text":"s^2 plus 1 doesn\u0027t have any roots, can\u0027t be factored."},{"Start":"00:24.740 ","End":"00:29.149","Text":"The general shape of the partial fraction is this."},{"Start":"00:29.149 ","End":"00:34.190","Text":"For the s, we need a constant over s and s^2 plus 1^2 we need to"},{"Start":"00:34.190 ","End":"00:39.875","Text":"take this thing squared unto the power of 1 and the linear term on the top for each."},{"Start":"00:39.875 ","End":"00:41.030","Text":"If you\u0027re not sure about this,"},{"Start":"00:41.030 ","End":"00:44.090","Text":"go and review the theory on partial fractions."},{"Start":"00:44.090 ","End":"00:49.925","Text":"I got rid of the denominators by multiplying by this denominator on both sides."},{"Start":"00:49.925 ","End":"00:52.520","Text":"Our aim of course, is to find the constants A, B,"},{"Start":"00:52.520 ","End":"00:55.570","Text":"C, D, E. Let\u0027s just simplify this a bit."},{"Start":"00:55.570 ","End":"00:58.725","Text":"Here I^2 the s^2 plus 1,"},{"Start":"00:58.725 ","End":"01:04.275","Text":"the s I threw inside this brackets and similarly here."},{"Start":"01:04.275 ","End":"01:09.065","Text":"By the way, the first step could\u0027ve been to let s=0."},{"Start":"01:09.065 ","End":"01:11.240","Text":"If we did that, this would have come out 0,"},{"Start":"01:11.240 ","End":"01:14.980","Text":"this would have come out 0 and we would have got a=1."},{"Start":"01:14.980 ","End":"01:17.450","Text":"We\u0027ll see this anyway, the other way."},{"Start":"01:17.450 ","End":"01:19.460","Text":"It means there\u0027s more than 1 way to solve this."},{"Start":"01:19.460 ","End":"01:22.130","Text":"Now we continue expanding and now"},{"Start":"01:22.130 ","End":"01:25.795","Text":"all the brackets have opened and now we want to start collecting like terms."},{"Start":"01:25.795 ","End":"01:28.185","Text":"According to increasing powers of s,"},{"Start":"01:28.185 ","End":"01:30.210","Text":"constants just s, s^2, s^3,"},{"Start":"01:30.210 ","End":"01:34.050","Text":"s^4 and what we\u0027re going to do is compare powers of s. I"},{"Start":"01:34.050 ","End":"01:38.345","Text":"have 2 polynomials on each side of the equals,"},{"Start":"01:38.345 ","End":"01:45.905","Text":"and I need to compare 2 polynomials are the same then all the coefficients are the same."},{"Start":"01:45.905 ","End":"01:47.210","Text":"The left-hand side is 1,"},{"Start":"01:47.210 ","End":"01:49.130","Text":"so the right-hand side has to be 1."},{"Start":"01:49.130 ","End":"01:52.145","Text":"All these coefficients are going to be 0,"},{"Start":"01:52.145 ","End":"01:54.070","Text":"but this one\u0027s going to be 1."},{"Start":"01:54.070 ","End":"01:55.470","Text":"Unlike we said before,"},{"Start":"01:55.470 ","End":"01:57.340","Text":"A comes out to be 1."},{"Start":"01:57.340 ","End":"02:00.800","Text":"Anyway, we also have C plus E is 0 and so on."},{"Start":"02:00.800 ","End":"02:02.030","Text":"All these coefficients are 0."},{"Start":"02:02.030 ","End":"02:05.150","Text":"We\u0027ve also found C already, it\u0027s 0."},{"Start":"02:05.150 ","End":"02:09.140","Text":"From here, we can solve all of them for example, or let\u0027s just do it all."},{"Start":"02:09.140 ","End":"02:11.255","Text":"A is 1, plug it in here."},{"Start":"02:11.255 ","End":"02:13.375","Text":"We\u0027ve got B is minus 1."},{"Start":"02:13.375 ","End":"02:16.585","Text":"Then we already said that A is 1, C is 0."},{"Start":"02:16.585 ","End":"02:19.940","Text":"E we can get from plugging C=0 here,"},{"Start":"02:19.940 ","End":"02:25.590","Text":"so that\u0027s also 0 and D we can get by plugging A and B both in here."},{"Start":"02:25.590 ","End":"02:27.780","Text":"A is 1, B is minus 1,"},{"Start":"02:27.780 ","End":"02:29.710","Text":"so we get D is minus 1."},{"Start":"02:29.710 ","End":"02:31.444","Text":"Now we have all the constants."},{"Start":"02:31.444 ","End":"02:34.580","Text":"So let\u0027s put them into the expression for"},{"Start":"02:34.580 ","End":"02:38.030","Text":"the partial fractions and we get this, this, and this."},{"Start":"02:38.030 ","End":"02:41.885","Text":"Of course, we are still looking for the inverse transform of this."},{"Start":"02:41.885 ","End":"02:45.889","Text":"This is just the partial fraction decomposition"},{"Start":"02:45.889 ","End":"02:49.580","Text":"of the original expression and this is easier to work with."},{"Start":"02:49.580 ","End":"02:54.590","Text":"Now we go and break it up further using linearity,"},{"Start":"02:54.590 ","End":"02:56.674","Text":"just break it up into pieces."},{"Start":"02:56.674 ","End":"03:02.795","Text":"For reference, I\u0027ve quoted the rule for linearity of the inverse transform."},{"Start":"03:02.795 ","End":"03:04.880","Text":"Here is written for 2 terms,"},{"Start":"03:04.880 ","End":"03:08.525","Text":"but it works for 3 terms or so in a similar way."},{"Start":"03:08.525 ","End":"03:11.600","Text":"It means that we just break each piece up when we can take constants like"},{"Start":"03:11.600 ","End":"03:15.255","Text":"minus 1 outside and we end up in this."},{"Start":"03:15.255 ","End":"03:19.610","Text":"At this point it\u0027s time to go and see what we have in our table of"},{"Start":"03:19.610 ","End":"03:23.990","Text":"inverse Laplace transforms and we find 3 rules that are going to help us."},{"Start":"03:23.990 ","End":"03:26.450","Text":"This one is this exactly,"},{"Start":"03:26.450 ","End":"03:28.130","Text":"this one is this,"},{"Start":"03:28.130 ","End":"03:32.180","Text":"but we have to put a=1 here."},{"Start":"03:32.180 ","End":"03:34.770","Text":"Here it looks like this,"},{"Start":"03:34.770 ","End":"03:39.410","Text":"but we are going to have to make it a little bit of adjustments."},{"Start":"03:39.410 ","End":"03:42.565","Text":"Because if I let a=1,"},{"Start":"03:42.565 ","End":"03:45.345","Text":"I don\u0027t get exactly this."},{"Start":"03:45.345 ","End":"03:48.760","Text":"I\u0027m going to get 2s instead of just s,"},{"Start":"03:48.760 ","End":"03:51.935","Text":"but we can easily make some adjustments."},{"Start":"03:51.935 ","End":"03:56.840","Text":"Anyway, the first term comes out to be just 1 and the second term"},{"Start":"03:56.840 ","End":"04:02.990","Text":"becomes minus cos t. We have cos at with a=1 here."},{"Start":"04:02.990 ","End":"04:04.430","Text":"So it\u0027s cos 1t,"},{"Start":"04:04.430 ","End":"04:11.690","Text":"which is just cos t. The last term in order to get it to be in this shape 2as a is 1."},{"Start":"04:11.690 ","End":"04:13.550","Text":"I just need to put a 2 here."},{"Start":"04:13.550 ","End":"04:17.180","Text":"But if I put a 2 here, I need to compensate so I put 1/2 here."},{"Start":"04:17.180 ","End":"04:22.055","Text":"Now I do have that this is equal to this with a=1."},{"Start":"04:22.055 ","End":"04:26.660","Text":"So we can finally get this tsin at is"},{"Start":"04:26.660 ","End":"04:32.160","Text":"this and the 1/2 from here and this is the final answer, so we\u0027re done."}],"ID":8029},{"Watched":false,"Name":"Exercise 28","Duration":"2m 32s","ChapterTopicVideoID":7957,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"This is a 2-part exercise."},{"Start":"00:03.930 ","End":"00:09.135","Text":"In part A, we have to compute the inverse Laplace transform of this expression."},{"Start":"00:09.135 ","End":"00:11.820","Text":"Afterwards, we want to express the result as"},{"Start":"00:11.820 ","End":"00:15.225","Text":"a piecewise function and to sketch its graph."},{"Start":"00:15.225 ","End":"00:17.940","Text":"Of course, we\u0027ll begin with part a."},{"Start":"00:17.940 ","End":"00:26.280","Text":"The first thing we\u0027ll do is split this up using the linearity of the inverse transform."},{"Start":"00:26.280 ","End":"00:30.240","Text":"For reference, this is the linearity rule when we have 2 terms,"},{"Start":"00:30.240 ","End":"00:33.068","Text":"but it works also for 3 terms."},{"Start":"00:33.068 ","End":"00:40.865","Text":"We break this up into this,3 terms and constants pulled out in front."},{"Start":"00:40.865 ","End":"00:44.936","Text":"Now I\u0027m going to slightly rewrite it."},{"Start":"00:44.936 ","End":"00:51.055","Text":"In both these terms I pulled the exponent in front."},{"Start":"00:51.055 ","End":"00:53.630","Text":"The reason I did this is because there\u0027s"},{"Start":"00:53.630 ","End":"00:58.517","Text":"a rule that tells me what to do when I have an exponent"},{"Start":"00:58.517 ","End":"01:08.730","Text":"times the function big F here whose inverse transform I know,"},{"Start":"01:08.730 ","End":"01:17.115","Text":"and this u is the unit step function."},{"Start":"01:17.115 ","End":"01:23.650","Text":"This thing comes out to be this 0 or 1 depending on where t is situated relative to k,"},{"Start":"01:23.650 ","End":"01:26.990","Text":"but we\u0027ll use this in part b of the exercise."},{"Start":"01:26.990 ","End":"01:33.110","Text":"I also use the lookup table to look up the basic building blocks,"},{"Start":"01:33.110 ","End":"01:35.000","Text":"the pieces that I\u0027m going to be using."},{"Start":"01:35.000 ","End":"01:36.935","Text":"Here, I\u0027m going to be using 1/s,"},{"Start":"01:36.935 ","End":"01:42.275","Text":"and here I\u0027m going be using 1/s^2."},{"Start":"01:42.275 ","End":"01:46.710","Text":"I have the inverse transforms of both of these,"},{"Start":"01:46.710 ","End":"01:52.140","Text":"and if I use these together with this rule,"},{"Start":"01:52.140 ","End":"01:54.255","Text":"then this is what I get."},{"Start":"01:54.255 ","End":"01:57.070","Text":"See, the 1/s gives me 1,"},{"Start":"01:57.070 ","End":"01:59.825","Text":"but here it\u0027s 3 times, so it\u0027s 3."},{"Start":"01:59.825 ","End":"02:02.360","Text":"Then I have the minus 4."},{"Start":"02:02.360 ","End":"02:09.500","Text":"Then I use this rule with k=1 here,"},{"Start":"02:09.500 ","End":"02:12.455","Text":"and here I use it with k=3,"},{"Start":"02:12.455 ","End":"02:17.930","Text":"so both of them have a coefficient of minus 4 and plus 4,"},{"Start":"02:17.930 ","End":"02:23.735","Text":"and just this expression with k being 1 or 3, this expression."},{"Start":"02:23.735 ","End":"02:33.220","Text":"Then part b, we\u0027ll do the expansion as a piecewise function. We\u0027re done for part a."}],"ID":8030},{"Watched":false,"Name":"Exercise 29","Duration":"2m 4s","ChapterTopicVideoID":7958,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.325","Text":"This is a 2 part exercise."},{"Start":"00:03.325 ","End":"00:08.500","Text":"In Part A, we have to compute the inverse Laplace transform of this expression."},{"Start":"00:08.500 ","End":"00:14.290","Text":"Afterwards we want to express the result as a piecewise function and to sketch its graph."},{"Start":"00:14.290 ","End":"00:16.430","Text":"Of course, we\u0027ll begin with Part A."},{"Start":"00:16.430 ","End":"00:22.795","Text":"The first thing we\u0027ll do is split this up using the linearity of the inverse transform."},{"Start":"00:22.795 ","End":"00:26.620","Text":"For reference, this is the linearity rule when we have 2 terms,"},{"Start":"00:26.620 ","End":"00:28.720","Text":"but it works also for 3 terms."},{"Start":"00:28.720 ","End":"00:35.080","Text":"We break this up into these 3 terms and constants pulled out in front."},{"Start":"00:35.080 ","End":"00:42.685","Text":"Now I\u0027m going to slightly rewrite it in both these terms I pulled the exponent in front."},{"Start":"00:42.685 ","End":"00:47.830","Text":"The reason I did this is because there\u0027s a rule that tells me what to do when I have an"},{"Start":"00:47.830 ","End":"00:55.600","Text":"exponent times a function F here whose inverse transform I know."},{"Start":"00:55.600 ","End":"00:59.480","Text":"This u is the unit step function."},{"Start":"00:59.480 ","End":"01:05.140","Text":"This thing comes out to be this 0 or 1 depending on where t is situated,"},{"Start":"01:05.140 ","End":"01:09.025","Text":"relative to k. But we\u0027ll use this in Part B of the exercise."},{"Start":"01:09.025 ","End":"01:14.290","Text":"I also use the lookup table to look up the basic building blocks,"},{"Start":"01:14.290 ","End":"01:16.180","Text":"the pieces that I\u0027m going to be using."},{"Start":"01:16.180 ","End":"01:18.375","Text":"Here I\u0027m going to be using 1 over s,"},{"Start":"01:18.375 ","End":"01:21.165","Text":"and here I\u0027m going to be using 1 over s^2."},{"Start":"01:21.165 ","End":"01:25.380","Text":"I have the inverse transforms of both of these."},{"Start":"01:25.380 ","End":"01:29.435","Text":"If I use these together with this rule,"},{"Start":"01:29.435 ","End":"01:31.430","Text":"then this is what I get."},{"Start":"01:31.430 ","End":"01:34.145","Text":"The 1 over s gives me 1,"},{"Start":"01:34.145 ","End":"01:36.680","Text":"but here it\u0027s 3 times, so it\u0027s 3."},{"Start":"01:36.680 ","End":"01:45.120","Text":"Then I have the minus 4 and then I use this rule with k=1 here."},{"Start":"01:45.120 ","End":"01:48.900","Text":"Here I use it with k=3."},{"Start":"01:48.900 ","End":"01:52.760","Text":"Both of them have a coefficient of minus 4 and minus 4."},{"Start":"01:52.760 ","End":"01:57.715","Text":"I\u0027ll just this expression with k being 1 or 3, this expression."},{"Start":"01:57.715 ","End":"02:02.240","Text":"Then Part B, we\u0027ll do the expansion as a piecewise function."},{"Start":"02:02.240 ","End":"02:04.800","Text":"We\u0027re done for Part A."}],"ID":8031},{"Watched":false,"Name":"Exercise 30","Duration":"2m 22s","ChapterTopicVideoID":7959,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this exercise, we\u0027re going to compute the inverse"},{"Start":"00:02.910 ","End":"00:08.100","Text":"Laplace transform of this expression. Just copied it."},{"Start":"00:08.100 ","End":"00:13.280","Text":"Notice that there are exponents here and here, multiplicative exponents."},{"Start":"00:13.280 ","End":"00:16.250","Text":"I\u0027m going to rewrite this, not just yet,"},{"Start":"00:16.250 ","End":"00:19.395","Text":"in a moment, but I will be taking the exponents to the side."},{"Start":"00:19.395 ","End":"00:21.840","Text":"I want to show you what my motivation is."},{"Start":"00:21.840 ","End":"00:27.780","Text":"There\u0027s a rule for computing the inverse Laplace transform of an exponent times"},{"Start":"00:27.780 ","End":"00:34.290","Text":"the known function and it\u0027s equal to this where f is the transform of F and,"},{"Start":"00:34.290 ","End":"00:37.410","Text":"of course, u is the unit step function,"},{"Start":"00:37.410 ","End":"00:39.630","Text":"but here it shifted k to the right."},{"Start":"00:39.630 ","End":"00:41.415","Text":"This is its definition."},{"Start":"00:41.415 ","End":"00:43.310","Text":"After I rewrite this,"},{"Start":"00:43.310 ","End":"00:45.725","Text":"taking the exponents to the sides,"},{"Start":"00:45.725 ","End":"00:50.270","Text":"what I\u0027ll have is 1 over s plus 1 and 1 over s^2 plus 1."},{"Start":"00:50.270 ","End":"00:55.035","Text":"I\u0027ll also need to look up the inverse Laplace transform of those,"},{"Start":"00:55.035 ","End":"00:58.655","Text":"and in the table, I find that this give me this and this gives me this."},{"Start":"00:58.655 ","End":"01:03.350","Text":"Although in the table, it\u0027s probably written as 1 over s plus a and 1"},{"Start":"01:03.350 ","End":"01:08.495","Text":"over s^2 plus a^2 and a will turn out to be 1."},{"Start":"01:08.495 ","End":"01:13.560","Text":"It\u0027s e^minus at and maybe sine at or something similar to that."},{"Start":"01:13.560 ","End":"01:15.675","Text":"It may be 1 over a sine at."},{"Start":"01:15.675 ","End":"01:17.890","Text":"Let me do the rewriting I said."},{"Start":"01:17.890 ","End":"01:20.780","Text":"From here, I just put the e^minus 4 aside,"},{"Start":"01:20.780 ","End":"01:23.855","Text":"nothing major times this function."},{"Start":"01:23.855 ","End":"01:27.545","Text":"Here I write e^2s at the side times this function."},{"Start":"01:27.545 ","End":"01:30.125","Text":"Now, I can apply this rule twice."},{"Start":"01:30.125 ","End":"01:34.790","Text":"Once with k=4,"},{"Start":"01:34.790 ","End":"01:37.330","Text":"and once with k equals,"},{"Start":"01:37.330 ","End":"01:39.530","Text":"because this is plus, and this is a minus,"},{"Start":"01:39.530 ","End":"01:42.455","Text":"so k is going to be minus 2."},{"Start":"01:42.455 ","End":"01:45.335","Text":"Together with these look ups,"},{"Start":"01:45.335 ","End":"01:54.590","Text":"we will get this because this is u(t minus 4) and this is what we have here."},{"Start":"01:54.590 ","End":"01:56.885","Text":"If instead of t,"},{"Start":"01:56.885 ","End":"01:59.650","Text":"I put t minus 4,"},{"Start":"01:59.650 ","End":"02:02.610","Text":"then this will give me this."},{"Start":"02:02.610 ","End":"02:11.100","Text":"Similarly, if I take this sine t and replace t by t plus 2,"},{"Start":"02:11.100 ","End":"02:13.650","Text":"then this is what we get."},{"Start":"02:13.650 ","End":"02:19.550","Text":"This is the answer in terms of the unit step function and the exponent on the sine."},{"Start":"02:19.550 ","End":"02:22.290","Text":"Anyway, that\u0027s the answer."}],"ID":8032},{"Watched":false,"Name":"Exercise 31","Duration":"2m 31s","ChapterTopicVideoID":7960,"CourseChapterTopicPlaylistID":4254,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"Here we need to compute the Inverse Laplace transform of this thing,"},{"Start":"00:04.650 ","End":"00:07.290","Text":"which is quite complex really,"},{"Start":"00:07.290 ","End":"00:13.845","Text":"but we can break it down and I\u0027m going to use the following rule just copy it first here."},{"Start":"00:13.845 ","End":"00:17.625","Text":"Now the rule involves multiplying something by an exponent,"},{"Start":"00:17.625 ","End":"00:19.300","Text":"the rule is this and we\u0027ve seen it before,"},{"Start":"00:19.300 ","End":"00:23.925","Text":"so I won\u0027t go into great detail and notice e^minus 10 as e^minus ks."},{"Start":"00:23.925 ","End":"00:30.420","Text":"We\u0027re going to use this rule with k=10 and just as a reminder,"},{"Start":"00:30.420 ","End":"00:33.704","Text":"I know you know this, but the unit step function,"},{"Start":"00:33.704 ","End":"00:37.410","Text":"which is shifted k units to the right is this,"},{"Start":"00:37.410 ","End":"00:38.990","Text":"to find one way up to k,"},{"Start":"00:38.990 ","End":"00:40.610","Text":"another way from k,"},{"Start":"00:40.610 ","End":"00:43.790","Text":"jumps at k. In a moment I\u0027m going to rewrite this"},{"Start":"00:43.790 ","End":"00:47.044","Text":"and put it as e^minus 10s times something,"},{"Start":"00:47.044 ","End":"00:50.540","Text":"and that something will be 1/s minus 1s minus 2,"},{"Start":"00:50.540 ","End":"00:53.300","Text":"and I\u0027ll need the inverse transform of that,"},{"Start":"00:53.300 ","End":"00:55.555","Text":"that\u0027ll be my F(s)."},{"Start":"00:55.555 ","End":"01:00.455","Text":"This is not an all the tables and the condensed Laplace tables transform,"},{"Start":"01:00.455 ","End":"01:01.940","Text":"we don\u0027t always have this,"},{"Start":"01:01.940 ","End":"01:07.610","Text":"but in my table I have this rule for inverse Laplace transforms and in our case,"},{"Start":"01:07.610 ","End":"01:12.420","Text":"a is 1 and b is 2,"},{"Start":"01:12.420 ","End":"01:15.510","Text":"which will give us this and of course the 2 minus 1 is"},{"Start":"01:15.510 ","End":"01:18.760","Text":"1 and we\u0027ll be able to throw that out in a moment."},{"Start":"01:18.760 ","End":"01:23.510","Text":"Now like I said that I\u0027m rewriting the original expression as e to the minus"},{"Start":"01:23.510 ","End":"01:28.850","Text":"something s separately and this bit separately and then when I apply the rule,"},{"Start":"01:28.850 ","End":"01:37.400","Text":"this bit is F(s) and of course f is the transform of F. This is the answer,"},{"Start":"01:37.400 ","End":"01:39.890","Text":"but this needs a bit of explaining."},{"Start":"01:39.890 ","End":"01:43.110","Text":"The u of t minus 10 is clear,"},{"Start":"01:43.110 ","End":"01:45.900","Text":"it\u0027s just this with k=10,"},{"Start":"01:45.900 ","End":"01:48.120","Text":"but what is this?"},{"Start":"01:48.120 ","End":"01:52.380","Text":"Well, f(t) without the minus k is just this,"},{"Start":"01:52.380 ","End":"01:53.970","Text":"so I could say that f(t),"},{"Start":"01:53.970 ","End":"01:58.515","Text":"it should have just thrown out this 2 minus 1 is e to the t,"},{"Start":"01:58.515 ","End":"02:00.705","Text":"minus e to the 2t,"},{"Start":"02:00.705 ","End":"02:08.150","Text":"but f(t) minus 10 is this thing above."},{"Start":"02:08.150 ","End":"02:11.140","Text":"I\u0027ll just copy it e to the t minus a to the 2t."},{"Start":"02:11.140 ","End":"02:19.115","Text":"But when I replace t by t minus 10,"},{"Start":"02:19.115 ","End":"02:23.570","Text":"then t is t minus 10 and 2t is twice t minus 10,"},{"Start":"02:23.570 ","End":"02:25.640","Text":"so this is what we get."},{"Start":"02:25.640 ","End":"02:27.680","Text":"We get possibly simplify this,"},{"Start":"02:27.680 ","End":"02:29.930","Text":"but we don\u0027t need to,"},{"Start":"02:29.930 ","End":"02:32.370","Text":"this is the answer."}],"ID":8033}],"Thumbnail":null,"ID":4254},{"Name":"Solving ODEs with the Laplace Transform","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Solving ODEs with the Laplace Transform I","Duration":"6m 29s","ChapterTopicVideoID":7961,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/7961.jpeg","UploadDate":"2018-10-23T20:25:32.4670000","DurationForVideoObject":"PT6M29S","Description":null,"MetaTitle":"Solving ODEs with the Laplace Transform I: Video + Workbook | Proprep","MetaDescription":"The Laplace Transform - Solving ODEs with the Laplace Transform. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/ordinary-differential-equations/the-laplace-transform/solving-odes-with-the-laplace-transform/vid8034","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"A new topic in ODEs this time,"},{"Start":"00:03.330 ","End":"00:09.015","Text":"how we\u0027re going to use the Laplace transform to solve ODEs which are linear,"},{"Start":"00:09.015 ","End":"00:12.060","Text":"non-homogeneous and with constant coefficients."},{"Start":"00:12.060 ","End":"00:13.680","Text":"Now we\u0027re going do it in general,"},{"Start":"00:13.680 ","End":"00:18.120","Text":"but we\u0027ll start with second order later we\u0027ll extend it to any order."},{"Start":"00:18.120 ","End":"00:20.790","Text":"How do we do second order ODEs?"},{"Start":"00:20.790 ","End":"00:22.575","Text":"In fact, what does one look like?"},{"Start":"00:22.575 ","End":"00:25.290","Text":"This is what it is. You can see it\u0027s second order."},{"Start":"00:25.290 ","End":"00:31.200","Text":"Check. Linear? Yeah. Non-homogeneous because of the g constant coefficients a,"},{"Start":"00:31.200 ","End":"00:33.360","Text":"b, and c are just numbers here."},{"Start":"00:33.360 ","End":"00:35.490","Text":"What do we do with this?"},{"Start":"00:35.490 ","End":"00:38.329","Text":"I forgot to mention we have initial conditions."},{"Start":"00:38.329 ","End":"00:42.116","Text":"We\u0027re given y at t equals naught,"},{"Start":"00:42.116 ","End":"00:44.600","Text":"and we\u0027re given y\u0027 when t is naught, this one,"},{"Start":"00:44.600 ","End":"00:46.940","Text":"I\u0027ll call y_0,"},{"Start":"00:46.940 ","End":"00:50.360","Text":"y_1 and these are supposed to be given."},{"Start":"00:50.360 ","End":"00:53.960","Text":"Now the word Laplace transform appears here,"},{"Start":"00:53.960 ","End":"00:55.640","Text":"so that\u0027s a hint of what we\u0027re going to do."},{"Start":"00:55.640 ","End":"00:58.910","Text":"We\u0027re going to take the Laplace transform of both sides of this equation."},{"Start":"00:58.910 ","End":"01:02.960","Text":"I\u0027ll do it systematically in steps and it will be a bit abstract."},{"Start":"01:02.960 ","End":"01:05.390","Text":"But afterwards we\u0027ll have a worked example."},{"Start":"01:05.390 ","End":"01:06.680","Text":"The first step is what I said,"},{"Start":"01:06.680 ","End":"01:11.045","Text":"just apply the Laplace transform to both sides. Now what do we get?"},{"Start":"01:11.045 ","End":"01:14.045","Text":"Just write the letter L on both sides,"},{"Start":"01:14.045 ","End":"01:16.610","Text":"curly L or straight L or whatever."},{"Start":"01:16.610 ","End":"01:19.455","Text":"Let\u0027s see how we do this."},{"Start":"01:19.455 ","End":"01:23.825","Text":"First, I\u0027m just going to present the result and then I\u0027ll give some explanations."},{"Start":"01:23.825 ","End":"01:27.320","Text":"Now, I claim that the right-hand side gives this,"},{"Start":"01:27.320 ","End":"01:30.175","Text":"left-hand side gives this how so?"},{"Start":"01:30.175 ","End":"01:35.240","Text":"Following the convention that when we have a transform of a small letter function in t,"},{"Start":"01:35.240 ","End":"01:38.660","Text":"we just call it the same letter but uppercase."},{"Start":"01:38.660 ","End":"01:40.745","Text":"So instead of t, we have a function of s,"},{"Start":"01:40.745 ","End":"01:42.005","Text":"that\u0027s the right-hand side."},{"Start":"01:42.005 ","End":"01:43.940","Text":"Now what about the left-hand side?"},{"Start":"01:43.940 ","End":"01:50.240","Text":"Well, big Y(s) is going to be the Laplace transform of little y(t)."},{"Start":"01:50.240 ","End":"01:51.620","Text":"In fact, I might even write that."},{"Start":"01:51.620 ","End":"01:53.900","Text":"Now, how did I get to this?"},{"Start":"01:53.900 ","End":"01:58.280","Text":"You can either just accept it as a formula or you don\u0027t have to know the reason,"},{"Start":"01:58.280 ","End":"02:00.025","Text":"but I\u0027d like to show you the reason."},{"Start":"02:00.025 ","End":"02:03.125","Text":"If you look at the table of Laplace transforms,"},{"Start":"02:03.125 ","End":"02:07.850","Text":"There\u0027s a rule for the transform of a derivative and of a second derivative."},{"Start":"02:07.850 ","End":"02:10.970","Text":"In fact, there\u0027s even a general rule for the nth derivative,"},{"Start":"02:10.970 ","End":"02:12.965","Text":"but we just need first second,"},{"Start":"02:12.965 ","End":"02:15.500","Text":"big F is the Laplace transform of little f,"},{"Start":"02:15.500 ","End":"02:16.805","Text":"as in the table."},{"Start":"02:16.805 ","End":"02:20.495","Text":"Replace f with the letter Y,"},{"Start":"02:20.495 ","End":"02:23.900","Text":"big F with the letter big Y."},{"Start":"02:23.900 ","End":"02:29.495","Text":"Substitute in here Laplace transform using linearity."},{"Start":"02:29.495 ","End":"02:32.255","Text":"I\u0027m not going to give all the details and all the steps."},{"Start":"02:32.255 ","End":"02:35.990","Text":"Then after simplification and collecting terms,"},{"Start":"02:35.990 ","End":"02:37.775","Text":"this is what we\u0027ll get,"},{"Start":"02:37.775 ","End":"02:39.545","Text":"is just basic algebra."},{"Start":"02:39.545 ","End":"02:42.440","Text":"As I said, you just can remember it as a formula."},{"Start":"02:42.440 ","End":"02:44.395","Text":"Onto the next step."},{"Start":"02:44.395 ","End":"02:50.530","Text":"In step 2, we take this equation and we extract big Y(s) by just"},{"Start":"02:50.530 ","End":"02:53.290","Text":"putting this and this on the other side with"},{"Start":"02:53.290 ","End":"02:57.040","Text":"a plus and then dividing by the coefficient of y(f)."},{"Start":"02:57.040 ","End":"02:58.585","Text":"Straightforward algebra."},{"Start":"02:58.585 ","End":"03:00.955","Text":"We\u0027ve got big Y(s)."},{"Start":"03:00.955 ","End":"03:05.035","Text":"Then the last step we want to get from big Y, back to little y."},{"Start":"03:05.035 ","End":"03:08.530","Text":"The way we do that is just to apply the inverse transform,"},{"Start":"03:08.530 ","End":"03:10.810","Text":"because big Y is the Laplace transform of little y."},{"Start":"03:10.810 ","End":"03:12.130","Text":"So now we do the reverse."},{"Start":"03:12.130 ","End":"03:16.750","Text":"Like so little y is just the inverse Laplace transform of this function"},{"Start":"03:16.750 ","End":"03:21.505","Text":"of s. All we need now because this is a bit abstract is an example."},{"Start":"03:21.505 ","End":"03:24.950","Text":"That\u0027s what I\u0027m going to do now and I\u0027m going to move to a new page."},{"Start":"03:24.950 ","End":"03:30.500","Text":"Now we come to the example of solving the ODE with Laplace transform."},{"Start":"03:30.500 ","End":"03:36.290","Text":"Here\u0027s the equation. It\u0027s second order constant coefficient, non-homogeneous, linear."},{"Start":"03:36.290 ","End":"03:38.240","Text":"We also have the initial conditions."},{"Start":"03:38.240 ","End":"03:41.275","Text":"We have y at zero and y\u0027 at zero."},{"Start":"03:41.275 ","End":"03:44.540","Text":"They\u0027re both zero. The first step is to"},{"Start":"03:44.540 ","End":"03:47.885","Text":"take the Laplace transform of both sides of the equation."},{"Start":"03:47.885 ","End":"03:52.640","Text":"First of all, I\u0027ll just write the letter L in front of them to indicate that I want"},{"Start":"03:52.640 ","End":"03:54.590","Text":"the Laplace transform and I\u0027ll"},{"Start":"03:54.590 ","End":"03:57.450","Text":"compute separately the transform of the left and on the right."},{"Start":"03:57.450 ","End":"03:58.955","Text":"On the left hand,"},{"Start":"03:58.955 ","End":"04:03.620","Text":"we recall the formula for how it is in general with second order equations."},{"Start":"04:03.620 ","End":"04:04.955","Text":"In our particular case,"},{"Start":"04:04.955 ","End":"04:07.685","Text":"a is 1, b is 5, c is 6."},{"Start":"04:07.685 ","End":"04:09.635","Text":"So if I substitute that,"},{"Start":"04:09.635 ","End":"04:13.390","Text":"we get here s^2 plus 5s plus 6."},{"Start":"04:13.390 ","End":"04:15.680","Text":"Here we have s plus 5."},{"Start":"04:15.680 ","End":"04:18.365","Text":"Here, just 1, y(0),"},{"Start":"04:18.365 ","End":"04:19.580","Text":"we already know is naught,"},{"Start":"04:19.580 ","End":"04:21.880","Text":"y\u0027(0) is also naught."},{"Start":"04:21.880 ","End":"04:23.750","Text":"Let\u0027s do the right-hand side."},{"Start":"04:23.750 ","End":"04:26.750","Text":"For the right-hand side, you want Laplace transform of 12."},{"Start":"04:26.750 ","End":"04:28.280","Text":"Now, it\u0027s not 12, the number."},{"Start":"04:28.280 ","End":"04:31.655","Text":"It\u0027s 12, the constant function that\u0027s always equal to 12."},{"Start":"04:31.655 ","End":"04:35.585","Text":"Now because the Laplace transform of 1 is 1 over s,"},{"Start":"04:35.585 ","End":"04:37.100","Text":"multiply it by 12,"},{"Start":"04:37.100 ","End":"04:39.800","Text":"we get 12 over s by linearity."},{"Start":"04:39.800 ","End":"04:44.270","Text":"Now, we can compare the left and right-hand sides."},{"Start":"04:44.270 ","End":"04:46.460","Text":"Here\u0027s the left, here\u0027s the right."},{"Start":"04:46.460 ","End":"04:47.840","Text":"We get, like I said,"},{"Start":"04:47.840 ","End":"04:49.850","Text":"this is 0 and 0, which makes it easy for us."},{"Start":"04:49.850 ","End":"04:52.760","Text":"We just have this for the left-hand side"},{"Start":"04:52.760 ","End":"04:56.165","Text":"and for the right-hand side we are 12 over s. So now we\u0027ve got this."},{"Start":"04:56.165 ","End":"04:58.175","Text":"Next, we\u0027re going to want to extract y."},{"Start":"04:58.175 ","End":"04:59.390","Text":"That\u0027s straightforward enough."},{"Start":"04:59.390 ","End":"05:02.270","Text":"Just bring this to the denominator of the other side."},{"Start":"05:02.270 ","End":"05:04.820","Text":"Now we have s. I also did an extra step."},{"Start":"05:04.820 ","End":"05:09.395","Text":"This thing factors into s plus 2, s plus 3."},{"Start":"05:09.395 ","End":"05:11.135","Text":"Check by multiplying out,"},{"Start":"05:11.135 ","End":"05:13.565","Text":"I think you know how to do this factoring."},{"Start":"05:13.565 ","End":"05:15.950","Text":"Lastly to get little y,"},{"Start":"05:15.950 ","End":"05:20.060","Text":"we just have to take the inverse transform of big Y, which is this."},{"Start":"05:20.060 ","End":"05:24.670","Text":"We need now to compute the inverse transform of this expression,"},{"Start":"05:24.670 ","End":"05:26.000","Text":"its a rational expression,"},{"Start":"05:26.000 ","End":"05:27.890","Text":"and we know how to do that."},{"Start":"05:27.890 ","End":"05:31.400","Text":"We\u0027ve done plenty of examples with partial fractions."},{"Start":"05:31.400 ","End":"05:33.080","Text":"Now here I\u0027m not going to give the details."},{"Start":"05:33.080 ","End":"05:34.190","Text":"We\u0027ve done plenty of those,"},{"Start":"05:34.190 ","End":"05:37.114","Text":"but if you break this expression up into partial fractions,"},{"Start":"05:37.114 ","End":"05:40.115","Text":"this is what we get and then we use linearity."},{"Start":"05:40.115 ","End":"05:42.950","Text":"I break it up into the separate Laplace transform of"},{"Start":"05:42.950 ","End":"05:46.205","Text":"each of these terms and also bring the constant out in front."},{"Start":"05:46.205 ","End":"05:50.075","Text":"So now we have to compute 3 separate Laplace transforms."},{"Start":"05:50.075 ","End":"05:53.390","Text":"This one\u0027s immediate and these two are almost immediate."},{"Start":"05:53.390 ","End":"05:56.690","Text":"The inverse transform of 1 over s you should know by now is 1."},{"Start":"05:56.690 ","End":"05:58.175","Text":"So here we have twice 1."},{"Start":"05:58.175 ","End":"06:00.620","Text":"Also, perhaps I\u0027ll write it,"},{"Start":"06:00.620 ","End":"06:10.880","Text":"the inverse Laplace transform of 1 over s plus a is e^minus at."},{"Start":"06:10.880 ","End":"06:14.165","Text":"If I let a equals 2 once,"},{"Start":"06:14.165 ","End":"06:17.375","Text":"and I let a equals 3 the second time,"},{"Start":"06:17.375 ","End":"06:22.655","Text":"what I get is e^minus 2t and e^minus 3t and the constant stay."},{"Start":"06:22.655 ","End":"06:24.920","Text":"This is the answer."},{"Start":"06:24.920 ","End":"06:30.360","Text":"In the next clip we\u0027ll be generalizing from second order to any order."}],"ID":8034},{"Watched":false,"Name":"Solving ODEs with the Laplace Transform II","Duration":"5m 1s","ChapterTopicVideoID":7962,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"This is a continuation of the previous clip,"},{"Start":"00:02.910 ","End":"00:04.590","Text":"which had the same title as this,"},{"Start":"00:04.590 ","End":"00:10.320","Text":"but it was second order and we\u0027re going to generalize from second order to any order."},{"Start":"00:10.320 ","End":"00:12.690","Text":"It\u0027s very similar to second order,"},{"Start":"00:12.690 ","End":"00:14.489","Text":"and the same general strategy."},{"Start":"00:14.489 ","End":"00:19.064","Text":"The main difference is the way we transform the left-hand side."},{"Start":"00:19.064 ","End":"00:21.135","Text":"In the case of first order,"},{"Start":"00:21.135 ","End":"00:23.265","Text":"it\u0027s even simpler than second order."},{"Start":"00:23.265 ","End":"00:31.960","Text":"The left-hand side is ay\u0027 plus by and the Laplace transform is this expression,"},{"Start":"00:31.960 ","End":"00:34.880","Text":"which is actually simpler than"},{"Start":"00:34.880 ","End":"00:39.245","Text":"the second order and has the same pattern but we\u0027ll talk about pattern in a moment."},{"Start":"00:39.245 ","End":"00:45.365","Text":"The second order is what we just had in the previous clip and its transform is this."},{"Start":"00:45.365 ","End":"00:49.310","Text":"You might already see a pattern between this and the previous one,"},{"Start":"00:49.310 ","End":"00:53.645","Text":"but let\u0027s go to third order and then see if we can get the general rule."},{"Start":"00:53.645 ","End":"00:55.340","Text":"In the case of third order,"},{"Start":"00:55.340 ","End":"00:59.375","Text":"the left-hand side of the differential equation looks like this."},{"Start":"00:59.375 ","End":"01:04.970","Text":"It starts with triple prime and we just label the coefficients alphabetically."},{"Start":"01:04.970 ","End":"01:09.215","Text":"The Laplace transform of this is what I\u0027ve written here."},{"Start":"01:09.215 ","End":"01:13.280","Text":"Let\u0027s see if we can see a pattern."},{"Start":"01:13.280 ","End":"01:14.930","Text":"Let\u0027s start with a second order."},{"Start":"01:14.930 ","End":"01:17.320","Text":"When we have an expression like this,"},{"Start":"01:17.320 ","End":"01:23.090","Text":"we called it once the characteristic polynomial, we get this."},{"Start":"01:23.090 ","End":"01:28.220","Text":"You just replace second order with s^2 first order derivative,"},{"Start":"01:28.220 ","End":"01:33.290","Text":"s. Just the function itself is like 1 and we just copy the coefficients a,"},{"Start":"01:33.290 ","End":"01:35.420","Text":"b, c. In front,"},{"Start":"01:35.420 ","End":"01:37.250","Text":"we put big Y."},{"Start":"01:37.250 ","End":"01:42.330","Text":"Well here we have a big Y and this is the only place we see this big Y,"},{"Start":"01:42.330 ","End":"01:46.125","Text":"the function of s. Then here and here,"},{"Start":"01:46.125 ","End":"01:52.460","Text":"it\u0027s always minuses and we have y(0) and y\u0027(0),"},{"Start":"01:52.460 ","End":"01:55.540","Text":"successive order derivatives at naught."},{"Start":"01:55.540 ","End":"01:58.100","Text":"The rest of it you can see the pattern."},{"Start":"01:58.100 ","End":"02:00.020","Text":"It\u0027s like shifted to the right instead of a,"},{"Start":"02:00.020 ","End":"02:01.490","Text":"b, c we just have a, b."},{"Start":"02:01.490 ","End":"02:05.840","Text":"You could say it\u0027s drop the last coefficient and divide by s, and we get this."},{"Start":"02:05.840 ","End":"02:08.915","Text":"Drop the last coefficient and divide by s, we get this."},{"Start":"02:08.915 ","End":"02:11.840","Text":"Well, let\u0027s see that this works also in third order."},{"Start":"02:11.840 ","End":"02:17.060","Text":"We have this equation and you don\u0027t have to remember the word characteristic polynomial,"},{"Start":"02:17.060 ","End":"02:19.710","Text":"but all it means is we copy the coefficients a,"},{"Start":"02:19.710 ","End":"02:21.600","Text":"b, c, and d. In the third order,"},{"Start":"02:21.600 ","End":"02:23.515","Text":"the derivative is s^3,"},{"Start":"02:23.515 ","End":"02:26.630","Text":"second order s^2, first order s, and so on."},{"Start":"02:26.630 ","End":"02:31.305","Text":"Here again it starts with Y(s)."},{"Start":"02:31.305 ","End":"02:34.090","Text":"I\u0027m missing a term, sorry."},{"Start":"02:34.090 ","End":"02:37.015","Text":"Big oops, there if you were trying to figure out the pattern."},{"Start":"02:37.015 ","End":"02:40.370","Text":"Anyway, we have the function at 0,"},{"Start":"02:40.370 ","End":"02:43.655","Text":"the first derivative, the second derivative."},{"Start":"02:43.655 ","End":"02:46.340","Text":"They keep going up the orders of the derivatives."},{"Start":"02:46.340 ","End":"02:49.160","Text":"It\u0027s like here we had a function and the derivative."},{"Start":"02:49.160 ","End":"02:53.550","Text":"Here if we drop the d and divide by s, we get this."},{"Start":"02:53.550 ","End":"02:55.905","Text":"We drop the c and divide by s, we get this."},{"Start":"02:55.905 ","End":"02:58.665","Text":"Drop the b and divide by s, we get a."},{"Start":"02:58.665 ","End":"03:02.840","Text":"It actually even works on the first order equation."},{"Start":"03:02.840 ","End":"03:05.163","Text":"Let\u0027s just go back a moment."},{"Start":"03:05.163 ","End":"03:07.210","Text":"You see the a and the b here."},{"Start":"03:07.210 ","End":"03:09.740","Text":"The characteristic polynomial is this."},{"Start":"03:09.740 ","End":"03:13.130","Text":"As before, a big Y and then the function."},{"Start":"03:13.130 ","End":"03:16.160","Text":"There are no derivatives because there\u0027s no room for it."},{"Start":"03:16.160 ","End":"03:21.195","Text":"I may explicitly drop the b and divide by s, we\u0027ve got the a."},{"Start":"03:21.195 ","End":"03:24.290","Text":"So it looks like the pattern is fairly clear."},{"Start":"03:24.290 ","End":"03:27.380","Text":"The second and third order really explain what\u0027s going"},{"Start":"03:27.380 ","End":"03:31.720","Text":"on but let\u0027s write a fourth order one."},{"Start":"03:31.720 ","End":"03:35.240","Text":"Here\u0027s our left-hand side of a fourth order equation."},{"Start":"03:35.240 ","End":"03:37.370","Text":"We have quadruple prime."},{"Start":"03:37.370 ","End":"03:38.810","Text":"Although if it gets too big,"},{"Start":"03:38.810 ","End":"03:41.815","Text":"we sometimes just would write a 4 in brackets."},{"Start":"03:41.815 ","End":"03:44.400","Text":"Some people also use Roman numerals,"},{"Start":"03:44.400 ","End":"03:46.110","Text":"anyway, that\u0027s besides the point."},{"Start":"03:46.110 ","End":"03:48.965","Text":"The Laplace transform of this,"},{"Start":"03:48.965 ","End":"03:51.035","Text":"you might be able to guess it."},{"Start":"03:51.035 ","End":"03:52.985","Text":"In fact, you probably could,"},{"Start":"03:52.985 ","End":"03:55.255","Text":"but if you can\u0027t, I\u0027ll show you."},{"Start":"03:55.255 ","End":"04:00.250","Text":"Wasn\u0027t the equation just the left-hand side,"},{"Start":"04:00.250 ","End":"04:03.770","Text":"but the characteristic polynomial for this, is this."},{"Start":"04:03.770 ","End":"04:05.240","Text":"Which is just a, b, c,"},{"Start":"04:05.240 ","End":"04:07.790","Text":"d. Fourth derivative is s^4,"},{"Start":"04:07.790 ","End":"04:09.605","Text":"third derivative of s^3, and so on."},{"Start":"04:09.605 ","End":"04:14.030","Text":"These pieces we have a big Y and then evaluations at 0,"},{"Start":"04:14.030 ","End":"04:18.762","Text":"these are the initial conditions, y(0), y\u0027(0)."},{"Start":"04:18.762 ","End":"04:20.795","Text":"Maybe I\u0027ll emphasize, it\u0027s always with a minus."},{"Start":"04:20.795 ","End":"04:25.670","Text":"It\u0027s a minus double prime and triple prime."},{"Start":"04:25.670 ","End":"04:28.220","Text":"The remaining bits are using the rule;"},{"Start":"04:28.220 ","End":"04:31.910","Text":"Drop the last coefficient and divide by s. Drop the d"},{"Start":"04:31.910 ","End":"04:35.720","Text":"and divide by s. Drop the c and divide by s. Drop"},{"Start":"04:35.720 ","End":"04:39.290","Text":"the b and divide by s. I\u0027m not going to write"},{"Start":"04:39.290 ","End":"04:43.510","Text":"something in general for order n. I\u0027m not even going to write order 5."},{"Start":"04:43.510 ","End":"04:46.240","Text":"I\u0027m not going to guarantee it,"},{"Start":"04:46.240 ","End":"04:48.490","Text":"but you\u0027re unlikely to even get a third order,"},{"Start":"04:48.490 ","End":"04:50.720","Text":"let alone a fourth order or something higher."},{"Start":"04:50.720 ","End":"04:53.760","Text":"So I\u0027ll settle for the words, and so on."},{"Start":"04:53.760 ","End":"04:55.925","Text":"That\u0027s all I want to say."},{"Start":"04:55.925 ","End":"04:57.680","Text":"Following this, there\u0027ll be lots of"},{"Start":"04:57.680 ","End":"05:01.620","Text":"solved examples and there you\u0027ll really learn how to do them."}],"ID":8035},{"Watched":false,"Name":"Exercise 1","Duration":"3m 48s","ChapterTopicVideoID":7963,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"We have here differential equation with initial condition,"},{"Start":"00:04.680 ","End":"00:08.295","Text":"and we\u0027re going to solve it using the Laplace transform."},{"Start":"00:08.295 ","End":"00:10.935","Text":"This is first-order, by the way,"},{"Start":"00:10.935 ","End":"00:12.465","Text":"we\u0027ll do it in three steps."},{"Start":"00:12.465 ","End":"00:17.040","Text":"Step 1 is to take the Laplace transform of both sides."},{"Start":"00:17.040 ","End":"00:18.270","Text":"Just as an aside,"},{"Start":"00:18.270 ","End":"00:20.955","Text":"sometimes I write it L,"},{"Start":"00:20.955 ","End":"00:26.100","Text":"like this for the transform and sometimes a curly L, it\u0027s equivalent."},{"Start":"00:26.100 ","End":"00:32.175","Text":"Now, we use the formula for the Laplace transform of such an expression."},{"Start":"00:32.175 ","End":"00:35.850","Text":"Here is the formula for first-order."},{"Start":"00:35.850 ","End":"00:38.255","Text":"We want to substitute."},{"Start":"00:38.255 ","End":"00:39.770","Text":"If we look at this equation,"},{"Start":"00:39.770 ","End":"00:41.180","Text":"we see that a is 1."},{"Start":"00:41.180 ","End":"00:45.065","Text":"I mean, it\u0027s like a 1 here and b is 4,"},{"Start":"00:45.065 ","End":"00:52.065","Text":"and also we have y(0) is 0 from here."},{"Start":"00:52.065 ","End":"00:54.960","Text":"Again, a is 1,"},{"Start":"00:54.960 ","End":"00:58.365","Text":"b is 4 and a is 1."},{"Start":"00:58.365 ","End":"01:04.360","Text":"What we also need is the transform of the right-hand side."},{"Start":"01:04.360 ","End":"01:06.470","Text":"In a moment really should explain this better."},{"Start":"01:06.470 ","End":"01:07.970","Text":"I don\u0027t know if necessary or not,"},{"Start":"01:07.970 ","End":"01:11.585","Text":"but you just copy this and make the substitutions."},{"Start":"01:11.585 ","End":"01:14.570","Text":"Maybe I didn\u0027t need to explain. It\u0027s pretty straightforward."},{"Start":"01:14.570 ","End":"01:17.390","Text":"The Laplace transform of the right-hand side,"},{"Start":"01:17.390 ","End":"01:19.550","Text":"we go to the table of Laplace transforms,"},{"Start":"01:19.550 ","End":"01:21.950","Text":"we don\u0027t find this exactly,"},{"Start":"01:21.950 ","End":"01:25.795","Text":"but we find in general e^at."},{"Start":"01:25.795 ","End":"01:29.830","Text":"If we let a= -3,"},{"Start":"01:29.830 ","End":"01:34.685","Text":"then we will get exactly what is written on the right-hand side."},{"Start":"01:34.685 ","End":"01:38.380","Text":"According to this, we get 1 over S+3."},{"Start":"01:38.380 ","End":"01:41.824","Text":"Note the plus because it\u0027s a minus minus."},{"Start":"01:41.824 ","End":"01:47.180","Text":"Now we equate, this was the left-hand side."},{"Start":"01:47.180 ","End":"01:52.430","Text":"Mathematics sometimes you write LHS and this was the right-hand side."},{"Start":"01:52.430 ","End":"01:55.240","Text":"Left-hand side equals right-hand side."},{"Start":"01:55.240 ","End":"01:57.750","Text":"Because of the 0 here,"},{"Start":"01:57.750 ","End":"02:01.440","Text":"we just get y(s)+(s+4) here this."},{"Start":"02:01.440 ","End":"02:05.360","Text":"Now we divide both sides by S+4. You know what?"},{"Start":"02:05.360 ","End":"02:08.420","Text":"I\u0027ll make that part of step 2."},{"Start":"02:08.420 ","End":"02:10.580","Text":"Here it is after I divide by S+4."},{"Start":"02:10.580 ","End":"02:18.065","Text":"Step 2 is to find the inverse Laplace transform of Y(s)."},{"Start":"02:18.065 ","End":"02:21.965","Text":"Find little y(t) using partial fractions."},{"Start":"02:21.965 ","End":"02:23.600","Text":"We get from here to here."},{"Start":"02:23.600 ","End":"02:25.910","Text":"I\u0027m not going to give the details,"},{"Start":"02:25.910 ","End":"02:28.280","Text":"you know how to do partial fractions."},{"Start":"02:28.280 ","End":"02:33.865","Text":"Now we need to get back to y using the inverse transform,"},{"Start":"02:33.865 ","End":"02:39.025","Text":"but that\u0027s already step 3 as we defined the steps."},{"Start":"02:39.025 ","End":"02:44.480","Text":"The way I do this is using the lookup table of Laplace transforms."},{"Start":"02:44.480 ","End":"02:51.715","Text":"We don\u0027t have an exact match for 1 over S+3 but we do have the inverse transform."},{"Start":"02:51.715 ","End":"02:53.690","Text":"There\u0027s a little typo here."},{"Start":"02:53.690 ","End":"02:56.015","Text":"There shouldn\u0027t be a -1 here."},{"Start":"02:56.015 ","End":"03:00.830","Text":"Laplace transform of this is this which we want."},{"Start":"03:00.830 ","End":"03:05.360","Text":"Although, I really want to have written it that\u0027s L inverse"},{"Start":"03:05.360 ","End":"03:11.340","Text":"Laplace transform of 1 over s+a is e^-at."},{"Start":"03:11.740 ","End":"03:16.250","Text":"Sometimes you only have one table and if you have the transform,"},{"Start":"03:16.250 ","End":"03:19.010","Text":"you have the inverse transform to get from here to here and back."},{"Start":"03:19.010 ","End":"03:24.120","Text":"Anyway, we let a=3 for the first piece,"},{"Start":"03:24.120 ","End":"03:28.530","Text":"and we let a=4 for the second piece."},{"Start":"03:28.530 ","End":"03:32.720","Text":"We also use linearity of the inverse transform meaning we can"},{"Start":"03:32.720 ","End":"03:36.770","Text":"take the inverse transform of this separately and this and subtract."},{"Start":"03:36.770 ","End":"03:39.500","Text":"In short, we end up with this as our answer."},{"Start":"03:39.500 ","End":"03:43.520","Text":"We let a=3, so here a=3, is this,"},{"Start":"03:43.520 ","End":"03:48.990","Text":"a= 4, is this and the minus is here. We\u0027re done."}],"ID":8036},{"Watched":false,"Name":"Exercise 2","Duration":"3m 40s","ChapterTopicVideoID":7964,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"We have here an initial value problem that"},{"Start":"00:03.000 ","End":"00:05.745","Text":"is a differential equation with initial conditions,"},{"Start":"00:05.745 ","End":"00:08.865","Text":"which we\u0027re going to solve using the Laplace transform."},{"Start":"00:08.865 ","End":"00:12.330","Text":"This is a second-order differential equation,"},{"Start":"00:12.330 ","End":"00:16.650","Text":"non homogeneous linear constant coefficients and in the first step,"},{"Start":"00:16.650 ","End":"00:20.525","Text":"we take the Laplace transform of both sides."},{"Start":"00:20.525 ","End":"00:22.025","Text":"For the left-hand side,"},{"Start":"00:22.025 ","End":"00:26.060","Text":"this is the formula that we\u0027re going to use and if we look at this,"},{"Start":"00:26.060 ","End":"00:28.750","Text":"we see that a is 1,"},{"Start":"00:28.750 ","End":"00:30.720","Text":"b is 4,"},{"Start":"00:30.720 ","End":"00:34.190","Text":"c is 4 and if we substitute that,"},{"Start":"00:34.190 ","End":"00:35.765","Text":"this is what we get."},{"Start":"00:35.765 ","End":"00:40.175","Text":"We also noticed that we have y(0) and y\u0027(0) and everything,"},{"Start":"00:40.175 ","End":"00:46.475","Text":"so we want to equate it to the Laplace transform of the right-hand side."},{"Start":"00:46.475 ","End":"00:48.185","Text":"For the right-hand side,"},{"Start":"00:48.185 ","End":"00:54.140","Text":"we are going to use this formula in the table of transforms and of course in our case,"},{"Start":"00:54.140 ","End":"00:58.215","Text":"we\u0027re going to let a equal -2,"},{"Start":"00:58.215 ","End":"01:04.445","Text":"and if we put that in and we also use linearity for the factor 10,"},{"Start":"01:04.445 ","End":"01:08.615","Text":"so we get 10 over s + 2 because it\u0027s minus a minus."},{"Start":"01:08.615 ","End":"01:10.355","Text":"Now if we equate,"},{"Start":"01:10.355 ","End":"01:13.280","Text":"this is the left-hand side transform,"},{"Start":"01:13.280 ","End":"01:15.470","Text":"this is the right-hand side,"},{"Start":"01:15.470 ","End":"01:22.234","Text":"and so after simplifying this a little bit, we get this."},{"Start":"01:22.234 ","End":"01:28.110","Text":"We move on to step 2 where we want to extract the function Y(s)."},{"Start":"01:28.110 ","End":"01:32.015","Text":"First we move the s to the other side and now we\u0027re going to divide by this,"},{"Start":"01:32.015 ","End":"01:33.980","Text":"and I divided each term separately,"},{"Start":"01:33.980 ","End":"01:36.720","Text":"so here we had over s + 2,"},{"Start":"01:36.720 ","End":"01:39.015","Text":"s + 2^2, so it\u0027s s + 2^3."},{"Start":"01:39.015 ","End":"01:42.705","Text":"Here s/s (s+2)2 with a minus."},{"Start":"01:42.705 ","End":"01:44.380","Text":"We now move to step 3,"},{"Start":"01:44.380 ","End":"01:48.025","Text":"which is to find the inverse transform of this,"},{"Start":"01:48.025 ","End":"01:51.620","Text":"and when we\u0027ve done that, we will have our function y."},{"Start":"01:51.620 ","End":"01:55.685","Text":"Now this inverse Laplace transform is a little bit tricky in the table,"},{"Start":"01:55.685 ","End":"01:57.005","Text":"at least in my table,"},{"Start":"01:57.005 ","End":"02:05.950","Text":"I find an entry for the inverse Laplace transform of 1/s plus a^n."},{"Start":"02:06.200 ","End":"02:13.385","Text":"I want to somehow manipulate this so it\u0027s in terms of this and here\u0027s how we go about it."},{"Start":"02:13.385 ","End":"02:16.550","Text":"Rewrite this s as s plus 2 - 2."},{"Start":"02:16.550 ","End":"02:18.200","Text":"Obviously it\u0027s the same thing."},{"Start":"02:18.200 ","End":"02:21.200","Text":"Now, break this up into 2 separate terms."},{"Start":"02:21.200 ","End":"02:24.065","Text":"This gives me this and this."},{"Start":"02:24.065 ","End":"02:27.205","Text":"Now here I can cancel s + 2,"},{"Start":"02:27.205 ","End":"02:29.340","Text":"and that gives me this."},{"Start":"02:29.340 ","End":"02:31.790","Text":"Now I could use the linearity of"},{"Start":"02:31.790 ","End":"02:35.915","Text":"the inverse transform and I break it up and take the constants out,"},{"Start":"02:35.915 ","End":"02:37.340","Text":"and this is what I now get."},{"Start":"02:37.340 ","End":"02:41.750","Text":"Now look each of these pieces for inverse transform of this form,"},{"Start":"02:41.750 ","End":"02:45.205","Text":"with n being 3, 1 or 2."},{"Start":"02:45.205 ","End":"02:48.485","Text":"I didn\u0027t tell you what the inverse transform is."},{"Start":"02:48.485 ","End":"02:54.470","Text":"This is the inverse transform of this."},{"Start":"02:54.470 ","End":"02:56.600","Text":"I apply it 3 times, like I said,"},{"Start":"02:56.600 ","End":"02:57.950","Text":"with n being 3,"},{"Start":"02:57.950 ","End":"02:59.650","Text":"1 or 2,"},{"Start":"02:59.650 ","End":"03:06.760","Text":"so this one is what you get if you put n=3, here, n=1."},{"Start":"03:06.760 ","End":"03:12.020","Text":"The last one is with n=2 and then a simplification."},{"Start":"03:12.020 ","End":"03:15.275","Text":"Take e^-2t outside the brackets,"},{"Start":"03:15.275 ","End":"03:18.815","Text":"10/2 factorial is 5."},{"Start":"03:18.815 ","End":"03:21.500","Text":"I switched the order around here,"},{"Start":"03:21.500 ","End":"03:23.540","Text":"so this is term I take next,"},{"Start":"03:23.540 ","End":"03:32.550","Text":"and I\u0027ve got -2 t. The 1 factorial is just 1 and here 0 factorial is also 1 remember,"},{"Start":"03:32.550 ","End":"03:34.200","Text":"and t^0 is 1,"},{"Start":"03:34.200 ","End":"03:41.470","Text":"so it\u0027s just a^-2t with a minus that\u0027s the -1 and this is the answer."}],"ID":8037},{"Watched":false,"Name":"Exercise 3","Duration":"2m 16s","ChapterTopicVideoID":7965,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"A second-order differential equation and I\u0027m going"},{"Start":"00:02.700 ","End":"00:05.805","Text":"to solve it with the Laplace transform."},{"Start":"00:05.805 ","End":"00:11.370","Text":"The first step is to take the Laplace transform of both sides of the equation."},{"Start":"00:11.370 ","End":"00:14.340","Text":"We\u0027ll start with the left-hand side and we"},{"Start":"00:14.340 ","End":"00:17.685","Text":"use the following formula which you\u0027ve already seen before."},{"Start":"00:17.685 ","End":"00:20.700","Text":"We substitute the known values and we\u0027ll get this."},{"Start":"00:20.700 ","End":"00:22.815","Text":"Let me just show you in more detail."},{"Start":"00:22.815 ","End":"00:27.090","Text":"We know that a is 1,"},{"Start":"00:27.090 ","End":"00:29.880","Text":"b is -4,"},{"Start":"00:29.880 ","End":"00:33.400","Text":"and c is 0. So a,1b is -4, c is 0."},{"Start":"00:38.720 ","End":"00:45.850","Text":"Also, we know that y\u0027(0) is -4 and y(0) is -1."},{"Start":"00:45.850 ","End":"00:48.485","Text":"Anyway, if you substitute all these numbers in, we get this."},{"Start":"00:48.485 ","End":"00:50.360","Text":"That\u0027s the left-hand side."},{"Start":"00:50.360 ","End":"00:54.380","Text":"The right-hand side is a constant function and as such,"},{"Start":"00:54.380 ","End":"00:59.315","Text":"its Laplace transform is that constant over s and that case 16/s."},{"Start":"00:59.315 ","End":"01:02.150","Text":"Now we\u0027ll compare the left-hand side and the right-hand side."},{"Start":"01:02.150 ","End":"01:06.330","Text":"In other words, this has got to equal this,"},{"Start":"01:06.330 ","End":"01:09.875","Text":"and so we get this equation."},{"Start":"01:09.875 ","End":"01:13.940","Text":"In the next step, we\u0027re going to extract y from it,"},{"Start":"01:13.940 ","End":"01:18.560","Text":"which we do by putting the s on the other side and then dividing by this."},{"Start":"01:18.560 ","End":"01:21.665","Text":"Here\u0027s s on the other side and now the division,"},{"Start":"01:21.665 ","End":"01:24.785","Text":"not yet, let\u0027s first give it to a common denominator on the right."},{"Start":"01:24.785 ","End":"01:30.410","Text":"Now we\u0027ll divide and this gives us Y. I try and simplify this."},{"Start":"01:30.410 ","End":"01:34.175","Text":"I note that 4-s and s-4 are very close."},{"Start":"01:34.175 ","End":"01:38.006","Text":"I can write this as the minus the negative of s-4"},{"Start":"01:38.006 ","End":"01:41.795","Text":"and then these two will cancel leaving us with this."},{"Start":"01:41.795 ","End":"01:44.630","Text":"I can split this up as follows."},{"Start":"01:44.630 ","End":"01:47.450","Text":"Now on to the next step in which we are going to find"},{"Start":"01:47.450 ","End":"01:52.280","Text":"y(t) simply by taking the inverse Laplace transform of this, i.e."},{"Start":"01:52.280 ","End":"01:54.470","Text":"this, and for this,"},{"Start":"01:54.470 ","End":"01:57.200","Text":"we\u0027ll use the linearity rule."},{"Start":"01:57.200 ","End":"01:59.690","Text":"This breaks up as follows."},{"Start":"01:59.690 ","End":"02:02.570","Text":"The 4 comes out, term separate and we"},{"Start":"02:02.570 ","End":"02:05.540","Text":"know the inverse transform of 1/s^2 or if you don\u0027t,"},{"Start":"02:05.540 ","End":"02:06.950","Text":"you can look it up in the table."},{"Start":"02:06.950 ","End":"02:13.091","Text":"This is t, 1/s is 1 and this gives us the final answer,"},{"Start":"02:13.091 ","End":"02:16.580","Text":"-4(t)-1, and we\u0027re done."}],"ID":8038},{"Watched":false,"Name":"Exercise 4","Duration":"2m 24s","ChapterTopicVideoID":7966,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.370","Text":"Here we have this differential equation with initial conditions"},{"Start":"00:04.370 ","End":"00:09.195","Text":"and we\u0027re going to use the method of the Laplace transform to solve it."},{"Start":"00:09.195 ","End":"00:14.655","Text":"We start off by taking the Laplace transform of the two sides of the equation,"},{"Start":"00:14.655 ","End":"00:18.030","Text":"like so and we do each side separately."},{"Start":"00:18.030 ","End":"00:19.230","Text":"For the left-hand side,"},{"Start":"00:19.230 ","End":"00:21.930","Text":"we use the formula for second-order equations,"},{"Start":"00:21.930 ","End":"00:23.370","Text":"and here it is."},{"Start":"00:23.370 ","End":"00:26.490","Text":"Then in our case we see that a is 1,"},{"Start":"00:26.490 ","End":"00:29.940","Text":"b is equal to 4,"},{"Start":"00:29.940 ","End":"00:32.805","Text":"and c is equal to 0."},{"Start":"00:32.805 ","End":"00:34.500","Text":"Also from the initial conditions,"},{"Start":"00:34.500 ","End":"00:36.825","Text":"this is 0 when this is 0."},{"Start":"00:36.825 ","End":"00:38.505","Text":"If you substitute everything,"},{"Start":"00:38.505 ","End":"00:40.410","Text":"we end up with this."},{"Start":"00:40.410 ","End":"00:41.870","Text":"That\u0027s the left-hand side."},{"Start":"00:41.870 ","End":"00:43.750","Text":"Now the right-hand side."},{"Start":"00:43.750 ","End":"00:46.580","Text":"These are the transforms we\u0027re going to need,"},{"Start":"00:46.580 ","End":"00:47.810","Text":"I looked them up in the table,"},{"Start":"00:47.810 ","End":"00:49.340","Text":"well actually, I knew them by heart."},{"Start":"00:49.340 ","End":"00:51.950","Text":"That if we have t gives us 1/s^2,"},{"Start":"00:51.950 ","End":"00:54.095","Text":"and if we have 1 that gives us 1/s,"},{"Start":"00:54.095 ","End":"00:56.840","Text":"and I\u0027m going to use the linear property,"},{"Start":"00:56.840 ","End":"00:58.985","Text":"linearity of the transform."},{"Start":"00:58.985 ","End":"01:02.330","Text":"We have 8 times this plus twice this."},{"Start":"01:02.330 ","End":"01:07.190","Text":"This is what we get when we Laplace transform the right-hand side."},{"Start":"01:07.190 ","End":"01:11.345","Text":"Now, this is the left-hand side."},{"Start":"01:11.345 ","End":"01:13.640","Text":"This is the right-hand side,"},{"Start":"01:13.640 ","End":"01:16.065","Text":"and I\u0027m going to compare the two of them."},{"Start":"01:16.065 ","End":"01:19.160","Text":"This is the equation we have."},{"Start":"01:19.160 ","End":"01:21.335","Text":"On to the next step,"},{"Start":"01:21.335 ","End":"01:23.750","Text":"we want to isolate big Y,"},{"Start":"01:23.750 ","End":"01:25.130","Text":"a bit of algebra here,"},{"Start":"01:25.130 ","End":"01:30.680","Text":"take s out of these brackets and put these on a common denominator."},{"Start":"01:30.680 ","End":"01:34.340","Text":"Now we can divide by s, s plus 4."},{"Start":"01:34.340 ","End":"01:37.010","Text":"The s with s^2 makes it s^3."},{"Start":"01:37.010 ","End":"01:39.715","Text":"Then I can cancel the s plus 4."},{"Start":"01:39.715 ","End":"01:43.110","Text":"We get big Y of s is 2/s^3."},{"Start":"01:43.110 ","End":"01:45.890","Text":"Lastly, what we want to do is apply"},{"Start":"01:45.890 ","End":"01:49.955","Text":"the inverse transform to big Y in order to get little y."},{"Start":"01:49.955 ","End":"01:55.880","Text":"We need the inverse transform of 2/s^3 and we go to the table of transforms,"},{"Start":"01:55.880 ","End":"01:57.760","Text":"we find this entry."},{"Start":"01:57.760 ","End":"02:01.840","Text":"This will be useful to us if n is equal to 3."},{"Start":"02:01.840 ","End":"02:04.460","Text":"But we don\u0027t have 1/s^3, we have 2/s^2."},{"Start":"02:04.460 ","End":"02:09.110","Text":"We\u0027ll use also linearity to multiply by 2 and so this is what we get."},{"Start":"02:09.110 ","End":"02:12.885","Text":"We get t^3 minus 1 is t^2,"},{"Start":"02:12.885 ","End":"02:15.585","Text":"3 minus 1 factorial, 2 factorial,"},{"Start":"02:15.585 ","End":"02:17.520","Text":"the 2 from here,"},{"Start":"02:17.520 ","End":"02:19.080","Text":"but 2 factorial is 2,"},{"Start":"02:19.080 ","End":"02:20.389","Text":"so these 2 cancel,"},{"Start":"02:20.389 ","End":"02:23.359","Text":"so the answer is just t^2."},{"Start":"02:23.359 ","End":"02:25.620","Text":"We\u0027re done."}],"ID":8039},{"Watched":false,"Name":"Exercise 5","Duration":"4m 48s","ChapterTopicVideoID":7967,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"Here, we have a differential equation with initial conditions,"},{"Start":"00:03.900 ","End":"00:07.230","Text":"and we\u0027re going to solve it using the Laplace transform."},{"Start":"00:07.230 ","End":"00:12.225","Text":"The first step is just to take the Laplace transform of both sides of this equation."},{"Start":"00:12.225 ","End":"00:14.970","Text":"Now we have to actually perform the Laplace transform."},{"Start":"00:14.970 ","End":"00:17.430","Text":"Let\u0027s do each side separately,"},{"Start":"00:17.430 ","End":"00:19.320","Text":"starting with the left hand side."},{"Start":"00:19.320 ","End":"00:23.400","Text":"This is the familiar general rule for the second order equations."},{"Start":"00:23.400 ","End":"00:26.025","Text":"In our case we have that a is 4,"},{"Start":"00:26.025 ","End":"00:27.450","Text":"b is minus 4."},{"Start":"00:27.450 ","End":"00:29.609","Text":"There is no c, I mean it\u0027s 0."},{"Start":"00:29.609 ","End":"00:34.875","Text":"We also know y of naught is a quarter and so is y\u0027 of naught."},{"Start":"00:34.875 ","End":"00:37.380","Text":"If we substitute these constants in,"},{"Start":"00:37.380 ","End":"00:38.790","Text":"this is what we get,"},{"Start":"00:38.790 ","End":"00:41.250","Text":"and in a moment, we\u0027ll cancel."},{"Start":"00:41.250 ","End":"00:43.590","Text":"Now this thing can be divided by 4,"},{"Start":"00:43.590 ","End":"00:46.155","Text":"and this can be divided by 4 in a moment."},{"Start":"00:46.155 ","End":"00:49.565","Text":"Let\u0027s just do the right hand side first."},{"Start":"00:49.565 ","End":"00:53.265","Text":"I just wrote the result that perhaps I should explain a bit."},{"Start":"00:53.265 ","End":"00:56.175","Text":"This is te to the 1t,"},{"Start":"00:56.175 ","End":"01:01.385","Text":"and there is a rule that if I have t to the n,"},{"Start":"01:01.385 ","End":"01:04.890","Text":"e to the a_t in general."},{"Start":"01:04.890 ","End":"01:07.460","Text":"Now take the Laplace transform of that,"},{"Start":"01:07.460 ","End":"01:17.585","Text":"what I get is n factorial over s minus a to the power of n+1."},{"Start":"01:17.585 ","End":"01:19.430","Text":"Now in our case,"},{"Start":"01:19.430 ","End":"01:21.445","Text":"a is 1,"},{"Start":"01:21.445 ","End":"01:24.770","Text":"and then the first term n is 1,"},{"Start":"01:24.770 ","End":"01:27.200","Text":"and then the second term there is no power of t,"},{"Start":"01:27.200 ","End":"01:28.940","Text":"in other words, n=0."},{"Start":"01:28.940 ","End":"01:32.705","Text":"If we substitute these values respectively,"},{"Start":"01:32.705 ","End":"01:35.195","Text":"here we get n is 1, a is 1,"},{"Start":"01:35.195 ","End":"01:37.340","Text":"and here n is 0, a is 1,"},{"Start":"01:37.340 ","End":"01:39.965","Text":"and from this formula, this is what we get."},{"Start":"01:39.965 ","End":"01:45.565","Text":"Now we\u0027ve got the Laplace transform of the left hand side,"},{"Start":"01:45.565 ","End":"01:50.245","Text":"that\u0027s this, and the right hand side is this."},{"Start":"01:50.245 ","End":"01:54.440","Text":"My next step is just to compare the right hand side to the left hand side."},{"Start":"01:54.440 ","End":"01:56.120","Text":"The right hand side I just copied,"},{"Start":"01:56.120 ","End":"01:58.730","Text":"the left hand side I copied,"},{"Start":"01:58.730 ","End":"02:05.465","Text":"but I also simplified because here if I take a quarter of 4s minus 4,"},{"Start":"02:05.465 ","End":"02:08.580","Text":"it\u0027s like minus s minus 1,"},{"Start":"02:08.580 ","End":"02:09.945","Text":"when the 4\u0027s cancel,"},{"Start":"02:09.945 ","End":"02:13.335","Text":"and this will be quarter with 4 is 1 minus 1."},{"Start":"02:13.335 ","End":"02:16.725","Text":"If you expand it, it\u0027s minus s plus 1 minus 1,"},{"Start":"02:16.725 ","End":"02:21.460","Text":"so you end up with just the minus s that\u0027s here."},{"Start":"02:22.000 ","End":"02:25.775","Text":"We can move on to the next step,"},{"Start":"02:25.775 ","End":"02:28.855","Text":"which is to isolate Y(s)."},{"Start":"02:28.855 ","End":"02:33.170","Text":"First of all, I\u0027m going to put these over a common denominator."},{"Start":"02:33.170 ","End":"02:34.955","Text":"I\u0027m just going to do some algebra."},{"Start":"02:34.955 ","End":"02:39.840","Text":"Multiply this top and bottom by s minus 1,"},{"Start":"02:39.840 ","End":"02:42.465","Text":"so I\u0027ve got 1, and then s minus 1,"},{"Start":"02:42.465 ","End":"02:45.360","Text":"and the 1 minus 1 cancels and we\u0027re left with this."},{"Start":"02:45.360 ","End":"02:47.620","Text":"Now I\u0027m going to bring this over, not yet,"},{"Start":"02:47.620 ","End":"02:50.930","Text":"just has an idea that I can take s out of the brackets"},{"Start":"02:50.930 ","End":"02:54.485","Text":"on the left hand side because I\u0027ve got s here and I\u0027ve got s here."},{"Start":"02:54.485 ","End":"02:57.365","Text":"If I take it out and put it in curly brackets,"},{"Start":"02:57.365 ","End":"03:00.320","Text":"I\u0027ve got this thing without this s,"},{"Start":"03:00.320 ","End":"03:02.450","Text":"and here it becomes minus 1."},{"Start":"03:02.450 ","End":"03:05.200","Text":"Now, see if I divide by s,"},{"Start":"03:05.200 ","End":"03:09.150","Text":"I get this, just divided this by s,"},{"Start":"03:09.150 ","End":"03:12.875","Text":"and I divided this s cancel with this."},{"Start":"03:12.875 ","End":"03:16.265","Text":"Now we\u0027re going to bring the 1 over,"},{"Start":"03:16.265 ","End":"03:20.675","Text":"and now we can extract Y as follows."},{"Start":"03:20.675 ","End":"03:23.595","Text":"I mean the 1 quarter is going to be on each of the terms."},{"Start":"03:23.595 ","End":"03:27.440","Text":"Then the s minus 1 in the denominator makes this 2 or 3,"},{"Start":"03:27.440 ","End":"03:29.225","Text":"so it\u0027s 1 over s minus 1^3."},{"Start":"03:29.225 ","End":"03:32.470","Text":"Here I just get the s minus 1 on the denominator,"},{"Start":"03:32.470 ","End":"03:34.845","Text":"so that\u0027s step 2."},{"Start":"03:34.845 ","End":"03:38.525","Text":"The last step is to compute the inverse transform of this."},{"Start":"03:38.525 ","End":"03:40.940","Text":"Inverse transform L to the minus 1."},{"Start":"03:40.940 ","End":"03:48.680","Text":"Now we go to table of inverse transforms and the entries that we find useful."},{"Start":"03:48.680 ","End":"03:53.675","Text":"Well, one entry but different values of n. We\u0027re going to use this formula,"},{"Start":"03:53.675 ","End":"03:56.540","Text":"which is exactly what we have here and here if"},{"Start":"03:56.540 ","End":"04:00.520","Text":"you leave the quarter out of it and one case n equals 3,"},{"Start":"04:00.520 ","End":"04:01.910","Text":"one case n equals 1,"},{"Start":"04:01.910 ","End":"04:04.130","Text":"and our a is equal to 1."},{"Start":"04:04.130 ","End":"04:07.700","Text":"I guess I didn\u0027t mention that, we\u0027re using linearity."},{"Start":"04:07.700 ","End":"04:10.100","Text":"We do it automatically after awhile."},{"Start":"04:10.100 ","End":"04:13.835","Text":"The quarter comes out in front and you apply L minus 1 to this bit,"},{"Start":"04:13.835 ","End":"04:16.835","Text":"and another quarter, and L minus 1 applied to this bit."},{"Start":"04:16.835 ","End":"04:20.840","Text":"We use the linearity property without stating it explicitly."},{"Start":"04:20.840 ","End":"04:24.230","Text":"We have the 1 quarter and then from this formula,"},{"Start":"04:24.230 ","End":"04:26.385","Text":"we get e to the 1t,"},{"Start":"04:26.385 ","End":"04:30.120","Text":"which is this, t to the 3 minus 1,"},{"Start":"04:30.120 ","End":"04:34.750","Text":"and 3 minus 1 factorial and the other 1 with n equals 1."},{"Start":"04:34.750 ","End":"04:39.005","Text":"This can be simplified to this because 2 factorial is 2 and 2 with 4 is 8,"},{"Start":"04:39.005 ","End":"04:42.935","Text":"so we take 1/8 but now we have to put an extra 2 here,"},{"Start":"04:42.935 ","End":"04:45.470","Text":"and also the e to the t comes out,"},{"Start":"04:45.470 ","End":"04:49.500","Text":"so this is the answer and we are done."}],"ID":8040},{"Watched":false,"Name":"Exercise 6","Duration":"5m 50s","ChapterTopicVideoID":7968,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"Here we have a differential equation with initial conditions,"},{"Start":"00:04.065 ","End":"00:10.320","Text":"and it uses the step function on the right-hand side of the equation,"},{"Start":"00:10.320 ","End":"00:13.890","Text":"and we \u0027re going to solve it using the Laplace transform."},{"Start":"00:13.890 ","End":"00:15.570","Text":"The first step, as always,"},{"Start":"00:15.570 ","End":"00:19.935","Text":"is to just take the Laplace transform of both sides of this equation."},{"Start":"00:19.935 ","End":"00:22.410","Text":"Let\u0027s start with the left-hand side."},{"Start":"00:22.410 ","End":"00:26.430","Text":"This is the formula we need for second-order equations,"},{"Start":"00:26.430 ","End":"00:29.010","Text":"and what we have is a,"},{"Start":"00:29.010 ","End":"00:33.840","Text":"b and c are 1 minus 3 and 2."},{"Start":"00:33.840 ","End":"00:35.355","Text":"What else do we need here?"},{"Start":"00:35.355 ","End":"00:36.810","Text":"We need y of node,"},{"Start":"00:36.810 ","End":"00:40.080","Text":"which is node because these are the initial conditions on here also,"},{"Start":"00:40.080 ","End":"00:44.660","Text":"and we get the transform of the left-hand side, is this."},{"Start":"00:44.660 ","End":"00:46.519","Text":"For the right-hand side,"},{"Start":"00:46.519 ","End":"00:48.095","Text":"we\u0027ll need a formula,"},{"Start":"00:48.095 ","End":"00:51.860","Text":"and we look in our formula sheet table of Laplace transforms,"},{"Start":"00:51.860 ","End":"00:53.075","Text":"and we find this,"},{"Start":"00:53.075 ","End":"00:55.655","Text":"which is very much like what we have,"},{"Start":"00:55.655 ","End":"00:59.155","Text":"except in our case that k=4,"},{"Start":"00:59.155 ","End":"01:00.860","Text":"and if we just plug that in,"},{"Start":"01:00.860 ","End":"01:04.760","Text":"we just copy this but with k=4 here and here."},{"Start":"01:04.760 ","End":"01:08.660","Text":"This is the Laplace transform of the left-hand side."},{"Start":"01:08.660 ","End":"01:11.785","Text":"This is the Laplace transform of the right-hand side,"},{"Start":"01:11.785 ","End":"01:15.315","Text":"and we want to equate these and here we are,"},{"Start":"01:15.315 ","End":"01:19.175","Text":"this here and this here."},{"Start":"01:19.175 ","End":"01:23.415","Text":"Then the next step we\u0027re going to extract big Y(s),"},{"Start":"01:23.415 ","End":"01:26.735","Text":"so we just bring this to the denominator."},{"Start":"01:26.735 ","End":"01:29.300","Text":"But, I\u0027d like to manipulate this"},{"Start":"01:29.300 ","End":"01:31.790","Text":"because whenever we\u0027re going to take the inverse Laplace transform,"},{"Start":"01:31.790 ","End":"01:33.320","Text":"so let\u0027s break it up a bit."},{"Start":"01:33.320 ","End":"01:39.395","Text":"The denominator here of this quadratic term factors into s minus 1 s minus 2,"},{"Start":"01:39.395 ","End":"01:40.700","Text":"many ways to do this,"},{"Start":"01:40.700 ","End":"01:45.260","Text":"one way is to find that the roots of this equals 0, 1, and 2."},{"Start":"01:45.260 ","End":"01:46.490","Text":"Anyway, however you do it,"},{"Start":"01:46.490 ","End":"01:47.950","Text":"this is what we get."},{"Start":"01:47.950 ","End":"01:50.660","Text":"Then we use partial fractions."},{"Start":"01:50.660 ","End":"01:53.930","Text":"Let\u0027s write that partial fractions,"},{"Start":"01:53.930 ","End":"01:56.675","Text":"and I\u0027m going to do this at the end."},{"Start":"01:56.675 ","End":"01:59.255","Text":"This something I owe you,"},{"Start":"01:59.255 ","End":"02:01.955","Text":"and I\u0027ll do it at the end and still want to break the flow."},{"Start":"02:01.955 ","End":"02:03.680","Text":"Now that we have big Y,"},{"Start":"02:03.680 ","End":"02:08.990","Text":"the last step will be to take the inverse transform of big Y(s) and get little y(t)."},{"Start":"02:08.990 ","End":"02:14.105","Text":"Here we are, little y is inverse transform of this I copied,"},{"Start":"02:14.105 ","End":"02:17.040","Text":"and for convenience, and for using a formula,"},{"Start":"02:17.040 ","End":"02:22.610","Text":"I\u0027m going to call this expression inside the inner brackets as big F(s),"},{"Start":"02:22.610 ","End":"02:27.290","Text":"and I did that because there is a formula for having a known function"},{"Start":"02:27.290 ","End":"02:32.374","Text":"multiplied by an exponent e to the something s, and in general,"},{"Start":"02:32.374 ","End":"02:35.180","Text":"it looks like this as written here,"},{"Start":"02:35.180 ","End":"02:37.100","Text":"where u is as usual,"},{"Start":"02:37.100 ","End":"02:42.320","Text":"the unit step function but shifted by k. Then our k is looking at this,"},{"Start":"02:42.320 ","End":"02:44.975","Text":"and this will want k=4,"},{"Start":"02:44.975 ","End":"02:47.510","Text":"and if we plug in k=4,"},{"Start":"02:47.510 ","End":"02:52.010","Text":"this is what will get and this is how we\u0027ll break this up of break it up,"},{"Start":"02:52.010 ","End":"02:53.800","Text":"but how we handle it."},{"Start":"02:53.800 ","End":"02:57.470","Text":"As always, little f would be the inverse transform of"},{"Start":"02:57.470 ","End":"03:00.875","Text":"big F. Use this big letter, small letter convention."},{"Start":"03:00.875 ","End":"03:03.155","Text":"Little f is the inverse transform,"},{"Start":"03:03.155 ","End":"03:09.325","Text":"and that\u0027s what I want to do now is find little f by taking inverse transform of this."},{"Start":"03:09.325 ","End":"03:13.200","Text":"The formula is I need of inverse transform of 1 over"},{"Start":"03:13.200 ","End":"03:17.200","Text":"s. This will serve me for s minus 1 and s minus 2,"},{"Start":"03:17.200 ","End":"03:19.130","Text":"which is what I had there to show you."},{"Start":"03:19.130 ","End":"03:20.840","Text":"Maybe I can fit something on."},{"Start":"03:20.840 ","End":"03:26.850","Text":"Here I had the 1 over s. I\u0027m going to use this and for this 1 over s minus 1,"},{"Start":"03:26.850 ","End":"03:32.740","Text":"and 1 over s minus 2 so at one time we\u0027re going to take a=1 and the other time a=2,"},{"Start":"03:32.740 ","End":"03:36.775","Text":"and of course we\u0027ll break this up using the linearity property."},{"Start":"03:36.775 ","End":"03:39.990","Text":"The first term was 0.5 over s,"},{"Start":"03:39.990 ","End":"03:42.320","Text":"that just gives us 0.5."},{"Start":"03:42.320 ","End":"03:46.950","Text":"Then we had minus and we had 1 over s minus 1."},{"Start":"03:46.950 ","End":"03:49.230","Text":"So if a is 1,"},{"Start":"03:49.230 ","End":"03:51.735","Text":"then we get e^1t,"},{"Start":"03:51.735 ","End":"03:53.760","Text":"put a 1 in here if you wanted,"},{"Start":"03:53.760 ","End":"03:57.835","Text":"and the next one was plus 1 over s minus 2, so it\u0027s e^2t."},{"Start":"03:57.835 ","End":"04:07.345","Text":"But, this formula says that what we want is u(t) minus 4, f(t) minus 4."},{"Start":"04:07.345 ","End":"04:09.750","Text":"Here\u0027s the u(t) minus 4,"},{"Start":"04:09.750 ","End":"04:12.320","Text":"and f(t) minus 4 I get from this,"},{"Start":"04:12.320 ","End":"04:19.100","Text":"I just replace t. I replace it with t minus 4,"},{"Start":"04:19.100 ","End":"04:21.230","Text":"and then it gives me this expression."},{"Start":"04:21.230 ","End":"04:23.345","Text":"This is f(t) minus 4."},{"Start":"04:23.345 ","End":"04:24.830","Text":"Now this is the answer,"},{"Start":"04:24.830 ","End":"04:29.750","Text":"but don\u0027t go yet because I still owe you the partial fractions."},{"Start":"04:29.750 ","End":"04:33.110","Text":"If you\u0027re comfortable with partial fractions and don\u0027t need it, then we\u0027re done."},{"Start":"04:33.110 ","End":"04:35.465","Text":"But if you want to see the partial fractions,"},{"Start":"04:35.465 ","End":"04:38.035","Text":"how I did it, I\u0027ll do that now."},{"Start":"04:38.035 ","End":"04:39.830","Text":"I\u0027ll Just briefly walk you through it."},{"Start":"04:39.830 ","End":"04:41.990","Text":"What we had was this expression,"},{"Start":"04:41.990 ","End":"04:43.730","Text":"and because they\u0027re all linear terms,"},{"Start":"04:43.730 ","End":"04:46.790","Text":"we use constants over each of these."},{"Start":"04:46.790 ","End":"04:47.960","Text":"So we\u0027ve got A, B,"},{"Start":"04:47.960 ","End":"04:50.180","Text":"and C that we need to find."},{"Start":"04:50.180 ","End":"04:53.150","Text":"Multiply both sides by the denominator,"},{"Start":"04:53.150 ","End":"04:54.680","Text":"and we get this,"},{"Start":"04:54.680 ","End":"04:57.170","Text":"and because all these are different linear terms,"},{"Start":"04:57.170 ","End":"05:00.350","Text":"we can make substitutions to find each variable."},{"Start":"05:00.350 ","End":"05:03.260","Text":"For example, if we let s=1,"},{"Start":"05:03.260 ","End":"05:05.955","Text":"this and this will be 0 and so on."},{"Start":"05:05.955 ","End":"05:09.000","Text":"Show it by letting s successively be 0,"},{"Start":"05:09.000 ","End":"05:12.665","Text":"then 1, then 2 will get each of the variables."},{"Start":"05:12.665 ","End":"05:14.989","Text":"Just for example, if s is 0,"},{"Start":"05:14.989 ","End":"05:17.570","Text":"this disappears and this disappears,"},{"Start":"05:17.570 ","End":"05:21.620","Text":"and we get A times minus 1 times minus 2 is 1,"},{"Start":"05:21.620 ","End":"05:24.085","Text":"so 2A is 1."},{"Start":"05:24.085 ","End":"05:29.540","Text":"I guess that gives us that A is 1/2,"},{"Start":"05:29.540 ","End":"05:32.975","Text":"but we wrote it as 0.5."},{"Start":"05:32.975 ","End":"05:36.340","Text":"Then here we get that B,"},{"Start":"05:36.340 ","End":"05:37.740","Text":"B times minus 1 is 1,"},{"Start":"05:37.740 ","End":"05:39.030","Text":"so B is minus 1,"},{"Start":"05:39.030 ","End":"05:41.880","Text":"and here we get C equals 1,"},{"Start":"05:41.880 ","End":"05:43.790","Text":"and that\u0027s what we got before,"},{"Start":"05:43.790 ","End":"05:45.350","Text":"if you go back and see."},{"Start":"05:45.350 ","End":"05:48.200","Text":"That was the asterisk I owed you,"},{"Start":"05:48.200 ","End":"05:50.910","Text":"and now we\u0027re really done."}],"ID":8041},{"Watched":false,"Name":"Exercise 7","Duration":"9m 12s","ChapterTopicVideoID":7969,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"Here, we have an initial value problem with a second-order differential equation,"},{"Start":"00:05.460 ","End":"00:07.170","Text":"and these are the initial conditions,"},{"Start":"00:07.170 ","End":"00:11.600","Text":"and f is defined piecewise as follows."},{"Start":"00:11.600 ","End":"00:16.675","Text":"It\u0027s 0 for t less than 1 and 2 for t greater or equal to 1."},{"Start":"00:16.675 ","End":"00:20.070","Text":"We\u0027re going to do it using the Laplace transform."},{"Start":"00:20.070 ","End":"00:22.140","Text":"The first step is to apply"},{"Start":"00:22.140 ","End":"00:26.985","Text":"the Laplace transform to the left and right-hand sides of this equation like so."},{"Start":"00:26.985 ","End":"00:28.740","Text":"For the left-hand side,"},{"Start":"00:28.740 ","End":"00:32.745","Text":"we have the standard formula for second degree equations."},{"Start":"00:32.745 ","End":"00:35.235","Text":"In our case a, b, and c,"},{"Start":"00:35.235 ","End":"00:38.790","Text":"we looked at this 1,"},{"Start":"00:38.790 ","End":"00:40.260","Text":"1, and 0."},{"Start":"00:40.260 ","End":"00:44.055","Text":"These constants from here, they\u0027re both 0."},{"Start":"00:44.055 ","End":"00:46.590","Text":"We have all the constants."},{"Start":"00:46.590 ","End":"00:50.120","Text":"We have the Laplace transform of the right-hand side is this."},{"Start":"00:50.120 ","End":"00:54.815","Text":"Let me just write left-hand side, sorry."},{"Start":"00:54.815 ","End":"00:57.365","Text":"Now we\u0027ll get to the right-hand side."},{"Start":"00:57.365 ","End":"00:59.930","Text":"There\u0027s more than 1 way to do this."},{"Start":"00:59.930 ","End":"01:03.605","Text":"I\u0027ll do it 1 way and at the end we\u0027ll do it another way."},{"Start":"01:03.605 ","End":"01:05.030","Text":"To compute the Laplace transform,"},{"Start":"01:05.030 ","End":"01:06.590","Text":"I\u0027ll just go straight to the definition of"},{"Start":"01:06.590 ","End":"01:09.340","Text":"the Laplace transform and not used lookup table."},{"Start":"01:09.340 ","End":"01:12.590","Text":"At the end I\u0027ll show you an alternative way of computing it."},{"Start":"01:12.590 ","End":"01:17.210","Text":"Like I said, we\u0027ll start off with computing it as an integral."},{"Start":"01:17.210 ","End":"01:20.540","Text":"The definition of the transform gives us an integral from 0"},{"Start":"01:20.540 ","End":"01:24.365","Text":"to infinity of our function times e to the minus st dt."},{"Start":"01:24.365 ","End":"01:27.770","Text":"This comes out to be a function s. I\u0027m going to need to"},{"Start":"01:27.770 ","End":"01:32.000","Text":"scroll back because I don\u0027t see my function f, here it is."},{"Start":"01:32.000 ","End":"01:36.665","Text":"Now I see that it has a definition which changes at t=1."},{"Start":"01:36.665 ","End":"01:38.570","Text":"I\u0027ll break this up into 2 integrals,"},{"Start":"01:38.570 ","End":"01:41.164","Text":"from 0-1 and from 1 to infinity."},{"Start":"01:41.164 ","End":"01:43.430","Text":"But from 0-1 it\u0027s 0."},{"Start":"01:43.430 ","End":"01:50.580","Text":"All I need to do is replace f(t) by 2 and take the integral from 1 to infinity, like so."},{"Start":"01:50.580 ","End":"01:54.005","Text":"This is a fairly easy integral to compute."},{"Start":"01:54.005 ","End":"01:57.210","Text":"Let\u0027s see. Now the integrals with respect to"},{"Start":"01:57.210 ","End":"02:00.620","Text":"t. S is like a constant and also this 2 is a constant."},{"Start":"02:00.620 ","End":"02:03.645","Text":"When we take the integral of e to the something t,"},{"Start":"02:03.645 ","End":"02:06.500","Text":"it\u0027s the same thing except we divide by minus s."},{"Start":"02:06.500 ","End":"02:09.514","Text":"That gives me the minus here and the s on the denominator."},{"Start":"02:09.514 ","End":"02:10.910","Text":"But it\u0027s a definite integral,"},{"Start":"02:10.910 ","End":"02:14.195","Text":"so I have to plug in the limits of integration."},{"Start":"02:14.195 ","End":"02:16.725","Text":"If I plug in 1,"},{"Start":"02:16.725 ","End":"02:19.475","Text":"let\u0027s do the infinity first."},{"Start":"02:19.475 ","End":"02:22.190","Text":"What happens is that e to the minus st,"},{"Start":"02:22.190 ","End":"02:24.290","Text":"when t goes to infinity,"},{"Start":"02:24.290 ","End":"02:26.270","Text":"is e to the minus infinity."},{"Start":"02:26.270 ","End":"02:30.800","Text":"We have to assume that s is bigger than 0 for that, well,"},{"Start":"02:30.800 ","End":"02:36.710","Text":"we can make these assumptions to restrict the domain of s. If s is positive,"},{"Start":"02:36.710 ","End":"02:39.590","Text":"then e to the minus something positive times infinity"},{"Start":"02:39.590 ","End":"02:42.295","Text":"is e to the minus infinity, which is 0."},{"Start":"02:42.295 ","End":"02:45.915","Text":"We\u0027ve got 0 for the infinity part."},{"Start":"02:45.915 ","End":"02:47.460","Text":"Then when t is 1,"},{"Start":"02:47.460 ","End":"02:48.570","Text":"we have to subtract it."},{"Start":"02:48.570 ","End":"02:52.880","Text":"We\u0027ve got, well, this is a constant minus 2/s,"},{"Start":"02:52.880 ","End":"02:56.750","Text":"e to the minus s and t is s times 1."},{"Start":"02:56.750 ","End":"02:59.190","Text":"Well, s times 1 is just s,"},{"Start":"02:59.190 ","End":"03:02.325","Text":"that\u0027s e to the minus s and a minus and minus is plus."},{"Start":"03:02.325 ","End":"03:06.815","Text":"This is the Laplace transform of little f,"},{"Start":"03:06.815 ","End":"03:09.049","Text":"which was the right-hand side."},{"Start":"03:09.049 ","End":"03:10.419","Text":"Let me write that,"},{"Start":"03:10.419 ","End":"03:13.595","Text":"that was the right-hand side."},{"Start":"03:13.595 ","End":"03:16.940","Text":"If we go back and see what the left side was,"},{"Start":"03:16.940 ","End":"03:21.300","Text":"you\u0027ll see that this was it and this is the right-hand side."},{"Start":"03:21.300 ","End":"03:22.620","Text":"The left equals the right,"},{"Start":"03:22.620 ","End":"03:26.880","Text":"and the next step is to extract big y."},{"Start":"03:26.880 ","End":"03:28.595","Text":"This is what we get."},{"Start":"03:28.595 ","End":"03:31.160","Text":"All I did was take the s^2 plus s from"},{"Start":"03:31.160 ","End":"03:34.145","Text":"the left and bring it to the denominator on the right."},{"Start":"03:34.145 ","End":"03:36.830","Text":"Now here in the denominator I can take s out of"},{"Start":"03:36.830 ","End":"03:39.710","Text":"this piece and it combines with this s to make s^2."},{"Start":"03:39.710 ","End":"03:45.410","Text":"We have s^2 times s plus 1 and the 2e to the minus s I pull out in front."},{"Start":"03:45.410 ","End":"03:49.235","Text":"I want to prepare big y for taking an inverse transform."},{"Start":"03:49.235 ","End":"03:50.930","Text":"From here to here,"},{"Start":"03:50.930 ","End":"03:54.440","Text":"I did it using partial fractions."},{"Start":"03:54.440 ","End":"03:57.950","Text":"I still owe you this and I\u0027ll do it at the end,"},{"Start":"03:57.950 ","End":"04:01.550","Text":"remind me to show you at the end if you\u0027re interested,"},{"Start":"04:01.550 ","End":"04:04.720","Text":"if you want to take my word for it, that\u0027s fine too."},{"Start":"04:04.720 ","End":"04:09.235","Text":"The last step is the inverse Laplace transform."},{"Start":"04:09.235 ","End":"04:11.742","Text":"I just wrote this inverse transform,"},{"Start":"04:11.742 ","End":"04:13.940","Text":"and this I copied."},{"Start":"04:13.940 ","End":"04:16.370","Text":"We\u0027re going to need some formulas,"},{"Start":"04:16.370 ","End":"04:19.675","Text":"use the linearity to pull the 2 out in front."},{"Start":"04:19.675 ","End":"04:23.455","Text":"Now this bit I\u0027m going to call F(s)."},{"Start":"04:23.455 ","End":"04:27.965","Text":"I\u0027m doing that because there is a rule that helps me to take"},{"Start":"04:27.965 ","End":"04:33.170","Text":"a function of s times e to the something s. That rule is this,"},{"Start":"04:33.170 ","End":"04:37.100","Text":"that if I take the inverse transform of e to the minus ks times a function,"},{"Start":"04:37.100 ","End":"04:43.205","Text":"f is the inverse transform of F. Then we"},{"Start":"04:43.205 ","End":"04:49.955","Text":"substitute t minus k and multiply by the shifted unit step function, which is this."},{"Start":"04:49.955 ","End":"04:51.415","Text":"In our case,"},{"Start":"04:51.415 ","End":"04:55.710","Text":"because this is e to the minus s is like e to the minus 1s,"},{"Start":"04:55.710 ","End":"04:58.520","Text":"so k is equal to 1 in our case."},{"Start":"04:58.520 ","End":"05:00.490","Text":"If I write k equals 1,"},{"Start":"05:00.490 ","End":"05:05.270","Text":"then this becomes this and what I still want to do is compute"},{"Start":"05:05.270 ","End":"05:10.400","Text":"f by taking the inverse transform of F, which is this."},{"Start":"05:10.400 ","End":"05:14.350","Text":"Let\u0027s see. I guess I have no choice but to scroll."},{"Start":"05:14.350 ","End":"05:17.225","Text":"These are the formulas I need."},{"Start":"05:17.225 ","End":"05:19.580","Text":"I need for 1/s,"},{"Start":"05:19.580 ","End":"05:21.240","Text":"the inverse transform is 1,"},{"Start":"05:21.240 ","End":"05:22.830","Text":"I need a 1 over s^2,"},{"Start":"05:22.830 ","End":"05:27.020","Text":"and I need a 1 over s plus 1 and I have the inverse transform of all of these."},{"Start":"05:27.020 ","End":"05:30.575","Text":"This is from a more general rule for 1 over s plus a."},{"Start":"05:30.575 ","End":"05:35.540","Text":"Anyway, this is what we get and I\u0027m going to have to apply"},{"Start":"05:35.540 ","End":"05:41.930","Text":"the linearity to this just to figure out f. Hang on, just off screen."},{"Start":"05:41.930 ","End":"05:44.674","Text":"But we had minus 1 of these,"},{"Start":"05:44.674 ","End":"05:48.545","Text":"plus 1 of these and plus 1 of these."},{"Start":"05:48.545 ","End":"05:52.160","Text":"What we get using these rules is"},{"Start":"05:52.160 ","End":"06:00.350","Text":"minus 1 plus t plus e to the minus t. But that\u0027s not what we want because it was shifted,"},{"Start":"06:00.350 ","End":"06:09.345","Text":"we had that extra e to the minus t. We need this u(t) minus 1, f(t) minus 1."},{"Start":"06:09.345 ","End":"06:11.190","Text":"Don\u0027t forget there was a 2 in front."},{"Start":"06:11.190 ","End":"06:18.120","Text":"Here\u0027s the 2, here\u0027s the u(t) minus 1 and f(t) minus 1 is what I get here."},{"Start":"06:18.120 ","End":"06:20.100","Text":"If I just put instead of t,"},{"Start":"06:20.100 ","End":"06:21.840","Text":"t minus 1,"},{"Start":"06:21.840 ","End":"06:23.180","Text":"minus 1 is the constant,"},{"Start":"06:23.180 ","End":"06:24.860","Text":"t becomes t minus 1."},{"Start":"06:24.860 ","End":"06:28.730","Text":"This becomes e to the minus brackets t minus 1."},{"Start":"06:28.730 ","End":"06:30.650","Text":"This is the final answer,"},{"Start":"06:30.650 ","End":"06:33.219","Text":"but we\u0027re not done because I still owe you about"},{"Start":"06:33.219 ","End":"06:36.395","Text":"the partial fractions and I was going to show you an alternative method"},{"Start":"06:36.395 ","End":"06:42.670","Text":"for finding the Laplace transform of f. The partial fraction which I just threw at you,"},{"Start":"06:42.670 ","End":"06:45.620","Text":"I\u0027ll just give you briefly how I got to my answer."},{"Start":"06:45.620 ","End":"06:47.760","Text":"We got, remember minus 1,"},{"Start":"06:47.760 ","End":"06:49.805","Text":"1, and 1. Let\u0027s see."},{"Start":"06:49.805 ","End":"06:53.585","Text":"What we do is multiply both sides by this denominator."},{"Start":"06:53.585 ","End":"06:55.190","Text":"Then we get this."},{"Start":"06:55.190 ","End":"06:57.920","Text":"Now we can get 2 of these values of a,"},{"Start":"06:57.920 ","End":"07:00.290","Text":"b, or c by substitution."},{"Start":"07:00.290 ","End":"07:03.470","Text":"If I let s equals 0 or x equals minus 1,"},{"Start":"07:03.470 ","End":"07:05.225","Text":"well, let\u0027s take 1 at a time."},{"Start":"07:05.225 ","End":"07:07.730","Text":"If s is 0, then this is 0,"},{"Start":"07:07.730 ","End":"07:10.250","Text":"and this last 1 is 0."},{"Start":"07:10.250 ","End":"07:14.840","Text":"I get b times 1 is 1, so b is 1."},{"Start":"07:14.840 ","End":"07:17.495","Text":"If I let s be minus 1,"},{"Start":"07:17.495 ","End":"07:19.040","Text":"then this is 0,"},{"Start":"07:19.040 ","End":"07:23.510","Text":"and this is 0, so I just have to put minus 1 into the last term."},{"Start":"07:23.510 ","End":"07:26.735","Text":"It\u0027s c times minus 1^2, which is just c,"},{"Start":"07:26.735 ","End":"07:31.565","Text":"is equal to 1. Now we still need a."},{"Start":"07:31.565 ","End":"07:35.000","Text":"What we do is plug in any value except the ones we\u0027ve had already."},{"Start":"07:35.000 ","End":"07:36.140","Text":"We had 0 and minus 1."},{"Start":"07:36.140 ","End":"07:37.610","Text":"Let\u0027s take s equals 1."},{"Start":"07:37.610 ","End":"07:40.745","Text":"Then I plug it into here and we get this expression."},{"Start":"07:40.745 ","End":"07:46.520","Text":"But we already have b and c. If you substitute b and c here and straight away,"},{"Start":"07:46.520 ","End":"07:48.920","Text":"we\u0027ll see that a is minus 1,"},{"Start":"07:48.920 ","End":"07:50.446","Text":"and that\u0027s what we had before,"},{"Start":"07:50.446 ","End":"07:52.100","Text":"and so that\u0027s that."},{"Start":"07:52.100 ","End":"07:57.070","Text":"I also said I\u0027ll show you a different way to do the Laplace transform."},{"Start":"07:57.070 ","End":"08:02.275","Text":"We have this piecewise defined f(t) written above."},{"Start":"08:02.275 ","End":"08:05.149","Text":"We did it from the definition using integration."},{"Start":"08:05.149 ","End":"08:06.835","Text":"There\u0027s another way to do it."},{"Start":"08:06.835 ","End":"08:08.677","Text":"Just recall what f was,"},{"Start":"08:08.677 ","End":"08:12.170","Text":"it was 0 for t that is less than 1,"},{"Start":"08:12.170 ","End":"08:16.160","Text":"and 2 when t is greater or equal to 1."},{"Start":"08:16.160 ","End":"08:20.555","Text":"Notice that I can take 2 outside the curly brace."},{"Start":"08:20.555 ","End":"08:23.105","Text":"Because if I divide each of these by 2,"},{"Start":"08:23.105 ","End":"08:24.390","Text":"0/2 is 0,"},{"Start":"08:24.390 ","End":"08:27.904","Text":"2/2 is 1, so it\u0027s twice this function."},{"Start":"08:27.904 ","End":"08:31.880","Text":"But this should look familiar to you because this is"},{"Start":"08:31.880 ","End":"08:39.310","Text":"just the unit step function that is shifted to start at 1 and so it\u0027s u(t) minus 1."},{"Start":"08:39.310 ","End":"08:42.630","Text":"Our f(t) is twice u(t) minus 1."},{"Start":"08:42.630 ","End":"08:45.125","Text":"Apply the Laplace transform to each."},{"Start":"08:45.125 ","End":"08:48.155","Text":"By linearity, I can bring the 2 out front."},{"Start":"08:48.155 ","End":"08:52.130","Text":"Now I look in my formula sheet table of Laplace transforms,"},{"Start":"08:52.130 ","End":"08:54.020","Text":"and I find this formula,"},{"Start":"08:54.020 ","End":"08:57.635","Text":"which is exactly what we need if k equals 1."},{"Start":"08:57.635 ","End":"09:03.265","Text":"The 2 stays and this becomes e to the minus 1s, I don\u0027t need the 1,"},{"Start":"09:03.265 ","End":"09:07.640","Text":"over s. That\u0027s the same answer as we got before for"},{"Start":"09:07.640 ","End":"09:12.900","Text":"the Laplace transform of this function f. That\u0027s it."}],"ID":8042},{"Watched":false,"Name":"Exercise 8","Duration":"5m 25s","ChapterTopicVideoID":7970,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"We have this differential equation to solve with initial conditions,"},{"Start":"00:04.875 ","End":"00:09.690","Text":"and the right-hand side h is defined piecewise as follows."},{"Start":"00:09.690 ","End":"00:14.730","Text":"We\u0027re going to use the Laplace transform to solve this second-order equation."},{"Start":"00:14.730 ","End":"00:18.800","Text":"What we do first is apply the Laplace transform to the left-hand side,"},{"Start":"00:18.800 ","End":"00:21.780","Text":"and to the right-hand side of the equation."},{"Start":"00:21.780 ","End":"00:23.505","Text":"Now there\u0027s a formula."},{"Start":"00:23.505 ","End":"00:26.160","Text":"Here it is, it\u0027s going to be useful for the left-hand side,"},{"Start":"00:26.160 ","End":"00:29.010","Text":"and in our case we know the constant a is 1,"},{"Start":"00:29.010 ","End":"00:31.875","Text":"we see that b is 5, c is 6."},{"Start":"00:31.875 ","End":"00:34.770","Text":"Also these constants, y of note,"},{"Start":"00:34.770 ","End":"00:40.625","Text":"and y prime of note they\u0027re both 0 and so after all the substitution, we get this."},{"Start":"00:40.625 ","End":"00:44.570","Text":"Now that was just the left-hand side this bit."},{"Start":"00:44.570 ","End":"00:47.990","Text":"Now we want to do the transform of h,"},{"Start":"00:47.990 ","End":"00:50.690","Text":"which we\u0027ll do according to the definition because you won\u0027t"},{"Start":"00:50.690 ","End":"00:54.355","Text":"find something strange like this in the formula sheet."},{"Start":"00:54.355 ","End":"01:01.520","Text":"The integral from 0 to infinity of h of t e to the minus st dt it\u0027s the standard formula."},{"Start":"01:01.520 ","End":"01:05.240","Text":"Let\u0027s see if we can see the function."},{"Start":"01:05.240 ","End":"01:06.835","Text":"Yeah, here it is."},{"Start":"01:06.835 ","End":"01:08.610","Text":"Between 0 and infinity,"},{"Start":"01:08.610 ","End":"01:12.970","Text":"the only place where it\u0027s non-zero is from 0 to 2."},{"Start":"01:12.970 ","End":"01:16.890","Text":"All I have to do is take the integral from 0 to 2 because outside of that is 0,"},{"Start":"01:16.890 ","End":"01:19.790","Text":"so it won\u0027t contribute anything to the integral."},{"Start":"01:19.790 ","End":"01:22.310","Text":"We get the integral from 0 to 2,"},{"Start":"01:22.310 ","End":"01:24.170","Text":"and from 0 to 2h of t is 1."},{"Start":"01:24.170 ","End":"01:27.185","Text":"We have 1 E to the minus st dt."},{"Start":"01:27.185 ","End":"01:30.175","Text":"Now this is an easy integral to perform,"},{"Start":"01:30.175 ","End":"01:33.645","Text":"don\u0027t forget with respect to t so S is a constant."},{"Start":"01:33.645 ","End":"01:37.640","Text":"The integral of this thing is just itself divided by minus s,"},{"Start":"01:37.640 ","End":"01:39.260","Text":"I put the minus here, the S here,"},{"Start":"01:39.260 ","End":"01:41.600","Text":"and we have to substitute limits of integration,"},{"Start":"01:41.600 ","End":"01:42.875","Text":"it\u0027s a definite integral."},{"Start":"01:42.875 ","End":"01:45.175","Text":"We put in 2 and 0 and subtract,"},{"Start":"01:45.175 ","End":"01:47.805","Text":"here we plug in t equals 2,"},{"Start":"01:47.805 ","End":"01:50.210","Text":"and here when t is 0,"},{"Start":"01:50.210 ","End":"01:51.905","Text":"this thing becomes just 1,"},{"Start":"01:51.905 ","End":"01:54.515","Text":"and it\u0027s a minus minus so it\u0027s a plus."},{"Start":"01:54.515 ","End":"01:56.173","Text":"We have the left-hand side,"},{"Start":"01:56.173 ","End":"01:58.140","Text":"and now we have the right-hand side."},{"Start":"01:58.140 ","End":"02:01.070","Text":"The next thing we do is equate the left-hand side to"},{"Start":"02:01.070 ","End":"02:06.170","Text":"the right-hand side and this is what we get in terms of big Y."},{"Start":"02:06.170 ","End":"02:11.390","Text":"Now, next step is to isolate big Y. I could just divide by this,"},{"Start":"02:11.390 ","End":"02:14.380","Text":"but I want to prepare it a bit because I know I\u0027m going to need"},{"Start":"02:14.380 ","End":"02:18.065","Text":"the inverse transform so I want to factorize this."},{"Start":"02:18.065 ","End":"02:19.745","Text":"This factor is as follows."},{"Start":"02:19.745 ","End":"02:21.710","Text":"You know how to do this, but you could"},{"Start":"02:21.710 ","End":"02:24.790","Text":"always solve the quadratic equation where this is 0,"},{"Start":"02:24.790 ","End":"02:27.145","Text":"you\u0027d find the roots of minus 2, and minus 3,"},{"Start":"02:27.145 ","End":"02:32.210","Text":"so s minus 1 root s minus the other root comes up plus plus and this is what we get."},{"Start":"02:32.210 ","End":"02:35.790","Text":"Now we\u0027ll divide and here\u0027s our function y."},{"Start":"02:35.790 ","End":"02:40.070","Text":"The next step will be to find the inverse Laplace transform."},{"Start":"02:40.070 ","End":"02:42.170","Text":"Now here I use linearity."},{"Start":"02:42.170 ","End":"02:43.940","Text":"We have the minus and the plus,"},{"Start":"02:43.940 ","End":"02:47.690","Text":"and we apply the Laplace transform separately to the first-term,"},{"Start":"02:47.690 ","End":"02:50.120","Text":"and to the second term. They\u0027re actually very similar."},{"Start":"02:50.120 ","End":"02:52.745","Text":"Notice that they have this common part, this bit here."},{"Start":"02:52.745 ","End":"02:56.360","Text":"It\u0027s also here but here it\u0027s got this extra exponent."},{"Start":"02:56.360 ","End":"03:01.970","Text":"What we want to do now is to split this up into partial fractions."},{"Start":"03:01.970 ","End":"03:06.050","Text":"However, this is rather tedious and we\u0027ve done so many of these,"},{"Start":"03:06.050 ","End":"03:10.955","Text":"so what I did is I just searched the Internet and I found that there actually is"},{"Start":"03:10.955 ","End":"03:16.623","Text":"an entry in the expanded Laplace transform table for this function."},{"Start":"03:16.623 ","End":"03:20.255","Text":"Here\u0027s the formula I found so we might as well use it."},{"Start":"03:20.255 ","End":"03:28.425","Text":"In our case, we\u0027re going to have that a is 2 and b is 3."},{"Start":"03:28.425 ","End":"03:30.065","Text":"If I substitute that,"},{"Start":"03:30.065 ","End":"03:35.175","Text":"we get the inverse transform of this thing is this."},{"Start":"03:35.175 ","End":"03:36.930","Text":"Just look 2b is 3,"},{"Start":"03:36.930 ","End":"03:38.310","Text":"2 times 3 is 6."},{"Start":"03:38.310 ","End":"03:40.950","Text":"Here it\u0027s 3 over 3 minus 2,"},{"Start":"03:40.950 ","End":"03:42.340","Text":"2 over 3 minus 2."},{"Start":"03:42.340 ","End":"03:45.020","Text":"You can see that the substitution gives this."},{"Start":"03:45.020 ","End":"03:46.280","Text":"I want to label them,"},{"Start":"03:46.280 ","End":"03:48.300","Text":"the function of s here,"},{"Start":"03:48.300 ","End":"03:54.110","Text":"I\u0027ll call it big F of s and what I got I\u0027ll call it little f of t. I can actually reuse"},{"Start":"03:54.110 ","End":"04:00.655","Text":"this result in here using the rule for multiplying a function by an exponent."},{"Start":"04:00.655 ","End":"04:07.805","Text":"What this general rule says that if I know the inverse transform of big F is little f,"},{"Start":"04:07.805 ","End":"04:11.090","Text":"then if I multiply by e to the minus ks,"},{"Start":"04:11.090 ","End":"04:13.040","Text":"so minus 2s here,"},{"Start":"04:13.040 ","End":"04:20.655","Text":"then the result will be u of t minus kf of t minus kyu is the step function."},{"Start":"04:20.655 ","End":"04:24.435","Text":"If it\u0027s minus k, it\u0027s shifted step function and it\u0027s defined like this."},{"Start":"04:24.435 ","End":"04:28.615","Text":"We want to let k equals 2."},{"Start":"04:28.615 ","End":"04:30.125","Text":"If we do that,"},{"Start":"04:30.125 ","End":"04:32.030","Text":"then this is the result we get."},{"Start":"04:32.030 ","End":"04:33.470","Text":"Let me explain."},{"Start":"04:33.470 ","End":"04:36.740","Text":"Little y was made up of two bits."},{"Start":"04:36.740 ","End":"04:40.070","Text":"The second bit we already found was"},{"Start":"04:40.070 ","End":"04:46.550","Text":"this 1 sixth something and that was our f of t and that\u0027s the bit here,"},{"Start":"04:46.550 ","End":"04:49.595","Text":"that\u0027s the f of t which I just copied from here."},{"Start":"04:49.595 ","End":"04:54.905","Text":"This whole thing is f of t. From this rule,"},{"Start":"04:54.905 ","End":"04:59.405","Text":"I need the u of t minus k,"},{"Start":"04:59.405 ","End":"05:01.520","Text":"which is this, this is the k, oh,"},{"Start":"05:01.520 ","End":"05:03.770","Text":"there was a minus here of course,"},{"Start":"05:03.770 ","End":"05:06.035","Text":"so that\u0027s the minus."},{"Start":"05:06.035 ","End":"05:08.675","Text":"This bit is what?"},{"Start":"05:08.675 ","End":"05:16.580","Text":"Actually this whole thing is f of t minus k or t minus 2 because if this is f of t,"},{"Start":"05:16.580 ","End":"05:19.400","Text":"f of t minus 2 means wherever I see t,"},{"Start":"05:19.400 ","End":"05:20.750","Text":"I put t minus 2,"},{"Start":"05:20.750 ","End":"05:22.040","Text":"that\u0027s in two places."},{"Start":"05:22.040 ","End":"05:26.430","Text":"That\u0027s here and that\u0027s here. We\u0027re done."}],"ID":8043},{"Watched":false,"Name":"Exercise 9","Duration":"5m 18s","ChapterTopicVideoID":7971,"CourseChapterTopicPlaylistID":4253,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Here, we have an initial value problem,"},{"Start":"00:03.330 ","End":"00:05.010","Text":"meaning a differential equation and"},{"Start":"00:05.010 ","End":"00:09.810","Text":"initial conditions and we\u0027re going to solve it with the Laplace transform."},{"Start":"00:09.810 ","End":"00:15.430","Text":"Notice that for the first time we have a third-order equation up to now you\u0027ve had first,"},{"Start":"00:15.430 ","End":"00:18.105","Text":"second order, now we have a third order."},{"Start":"00:18.105 ","End":"00:19.860","Text":"The method is still the same,"},{"Start":"00:19.860 ","End":"00:23.685","Text":"we take the Laplace transform of both sides, that\u0027s step one."},{"Start":"00:23.685 ","End":"00:29.460","Text":"Now just wrote the letter L to Laplace transform in front of each of these."},{"Start":"00:29.460 ","End":"00:32.790","Text":"Now for this, there\u0027s a formula, and here it is,"},{"Start":"00:32.790 ","End":"00:38.010","Text":"we discussed it in the tutorial and we have the coefficients a is 1,"},{"Start":"00:38.010 ","End":"00:39.585","Text":"b is 4,"},{"Start":"00:39.585 ","End":"00:41.145","Text":"I\u0027m reading off here,"},{"Start":"00:41.145 ","End":"00:44.800","Text":"c is 5 and d is 2."},{"Start":"00:44.800 ","End":"00:47.930","Text":"We\u0027re also going to need these coefficients,"},{"Start":"00:47.930 ","End":"00:52.519","Text":"y(naughts) and y\u0027(naught) both 0,"},{"Start":"00:52.519 ","End":"00:55.655","Text":"but y\u0027\u0027(naught) is 3."},{"Start":"00:55.655 ","End":"00:57.995","Text":"Now, if we substitute all these numbers,"},{"Start":"00:57.995 ","End":"01:04.359","Text":"then we get that the Laplace transform of the left-hand side is all this."},{"Start":"01:04.359 ","End":"01:09.890","Text":"I\u0027ll just write left-hand side and we\u0027ll simplify it later."},{"Start":"01:09.890 ","End":"01:14.495","Text":"Let\u0027s just go to the right-hand side where we have a cosine."},{"Start":"01:14.495 ","End":"01:18.940","Text":"Well, we take the 10 out first using linearity and then the cosine t,"},{"Start":"01:18.940 ","End":"01:20.720","Text":"you can look it up in the table."},{"Start":"01:20.720 ","End":"01:27.710","Text":"It actually appears in my table more generally that the Laplace transform of cosine"},{"Start":"01:27.710 ","End":"01:36.320","Text":"of at is s over s^2 plus a^2 and here we have a=1."},{"Start":"01:36.320 ","End":"01:38.245","Text":"This is what we get,"},{"Start":"01:38.245 ","End":"01:40.215","Text":"that\u0027s the left-hand side,"},{"Start":"01:40.215 ","End":"01:44.945","Text":"this is now the right-hand side and so I\u0027m going to compare these,"},{"Start":"01:44.945 ","End":"01:47.540","Text":"and this is what we get."},{"Start":"01:47.540 ","End":"01:51.770","Text":"I brought the 3 to the other side and put everything over a common denominator."},{"Start":"01:51.770 ","End":"01:52.985","Text":"The 3 gives me,"},{"Start":"01:52.985 ","End":"01:53.990","Text":"I multiply it by this,"},{"Start":"01:53.990 ","End":"01:56.465","Text":"the 3s^2 plus 3 and then the 10a stays."},{"Start":"01:56.465 ","End":"02:03.180","Text":"We\u0027ve got this and now we go onto the next step which is to isolate Y of"},{"Start":"02:03.180 ","End":"02:09.937","Text":"s. It should say here step two just divide both sides by this."},{"Start":"02:09.937 ","End":"02:16.915","Text":"This is now going into the denominator so I have that Y is this expression."},{"Start":"02:16.915 ","End":"02:20.660","Text":"I want to use partial fractions on this but for"},{"Start":"02:20.660 ","End":"02:24.350","Text":"that I need the denominator to be fully factorized."},{"Start":"02:24.350 ","End":"02:27.710","Text":"Now x^2 plus 1 is irreducible, has no roots,"},{"Start":"02:27.710 ","End":"02:32.509","Text":"but this is a cubic and we want to break this down."},{"Start":"02:32.509 ","End":"02:34.790","Text":"I want to find the roots of this polynomial,"},{"Start":"02:34.790 ","End":"02:39.529","Text":"so I set it to be equal to 0 and I\u0027m going to look first for whole number solutions."},{"Start":"02:39.529 ","End":"02:40.930","Text":"Now using the theorem,"},{"Start":"02:40.930 ","End":"02:42.890","Text":"there\u0027s a theorem on that which says that"},{"Start":"02:42.890 ","End":"02:46.175","Text":"a whole number solution has to divide the free co-efficient."},{"Start":"02:46.175 ","End":"02:49.715","Text":"The only devices of 2 are plus or minus 1 and plus or minus 2."},{"Start":"02:49.715 ","End":"02:52.105","Text":"We try plugging each of them in;"},{"Start":"02:52.105 ","End":"02:56.040","Text":"only two of them work minus 1 and minus 2."},{"Start":"02:56.040 ","End":"02:59.705","Text":"If we then try the derivative,"},{"Start":"02:59.705 ","End":"03:03.560","Text":"it turns out that minus 1 also satisfies the derivative equals 0,"},{"Start":"03:03.560 ","End":"03:05.004","Text":"so it\u0027s a double root."},{"Start":"03:05.004 ","End":"03:07.340","Text":"We have the roots is minus 1, minus 1,"},{"Start":"03:07.340 ","End":"03:10.400","Text":"and minus 2 and now I know how to factorize it."},{"Start":"03:10.400 ","End":"03:14.960","Text":"The s^2 plus 1 just stays and this is now s minus minus 1,"},{"Start":"03:14.960 ","End":"03:16.145","Text":"which is s plus 1,"},{"Start":"03:16.145 ","End":"03:20.950","Text":"but squared because it\u0027s double root and this one gives me the s plus 2."},{"Start":"03:20.950 ","End":"03:24.640","Text":"Clear some space and we wanted to do partial fractions."},{"Start":"03:24.640 ","End":"03:27.800","Text":"This is the general shape for the irreducible quadratic,"},{"Start":"03:27.800 ","End":"03:29.195","Text":"I need a linear term."},{"Start":"03:29.195 ","End":"03:33.530","Text":"The double root means I take it both to the power of 2 and to the power of 1."},{"Start":"03:33.530 ","End":"03:35.380","Text":"Here, just as is."},{"Start":"03:35.380 ","End":"03:37.010","Text":"We have 5 constants,"},{"Start":"03:37.010 ","End":"03:38.120","Text":"5 unknowns, A, B,"},{"Start":"03:38.120 ","End":"03:40.940","Text":"C, D, E. I\u0027m not going to go into the details,"},{"Start":"03:40.940 ","End":"03:43.820","Text":"I\u0027ll leave it to you as an exercise because it\u0027s quite"},{"Start":"03:43.820 ","End":"03:46.839","Text":"tedious and doesn\u0027t really teach us anything new."},{"Start":"03:46.839 ","End":"03:49.220","Text":"From here to here, partial fractions,"},{"Start":"03:49.220 ","End":"03:54.020","Text":"we multiply by this denominator and we assign different values."},{"Start":"03:54.020 ","End":"04:00.045","Text":"Anyway, these are the results we get for the five constants and plug them in here."},{"Start":"04:00.045 ","End":"04:06.110","Text":"Y(s) can be written like this."},{"Start":"04:06.110 ","End":"04:09.560","Text":"First thing I want to do is to break it further,"},{"Start":"04:09.560 ","End":"04:12.035","Text":"is to break this into 2 separate terms."},{"Start":"04:12.035 ","End":"04:18.110","Text":"I have a minus s and plus 2 and that gives me altogether,"},{"Start":"04:18.110 ","End":"04:19.260","Text":"now I have 5 terms,"},{"Start":"04:19.260 ","End":"04:20.660","Text":"this one is just copied here,"},{"Start":"04:20.660 ","End":"04:22.145","Text":"this one and this one."},{"Start":"04:22.145 ","End":"04:27.620","Text":"Then we\u0027re going to go to the last step which is to find the inverse transform of this."},{"Start":"04:27.620 ","End":"04:32.060","Text":"This is the step at which I introduced the linearity."},{"Start":"04:32.060 ","End":"04:38.210","Text":"Y(t) is the inverse transform of this but instead of just copying L^minus 1 of all this,"},{"Start":"04:38.210 ","End":"04:39.892","Text":"I broke it up separate."},{"Start":"04:39.892 ","End":"04:43.040","Text":"Take the constants out like the 2,2 and the minus 2,"},{"Start":"04:43.040 ","End":"04:44.750","Text":"here\u0027s the minus 1."},{"Start":"04:44.750 ","End":"04:47.285","Text":"These are the pieces I need."},{"Start":"04:47.285 ","End":"04:53.060","Text":"Then with the help of the lookup table for inverse Laplace transforms,"},{"Start":"04:53.060 ","End":"04:55.475","Text":"this one gives me cosine,"},{"Start":"04:55.475 ","End":"04:58.550","Text":"this bit gives me sine."},{"Start":"04:58.550 ","End":"05:01.550","Text":"Well, there\u0027s a formula for 1 over s plus a,"},{"Start":"05:01.550 ","End":"05:04.202","Text":"and then it\u0027s e^minus at, but a is 1."},{"Start":"05:04.202 ","End":"05:08.180","Text":"Similarly here there\u0027s a formula with a"},{"Start":"05:08.180 ","End":"05:12.650","Text":"but a is 1 and similarly here there\u0027s a formula 1 over s plus a,"},{"Start":"05:12.650 ","End":"05:13.895","Text":"but a is 2,"},{"Start":"05:13.895 ","End":"05:15.725","Text":"and this is what we get."},{"Start":"05:15.725 ","End":"05:18.870","Text":"Finally, we are done."}],"ID":8044}],"Thumbnail":null,"ID":4253}]

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