[{"Name":"Linear, Homogeneous, Constant Coefficients","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear, Homogeneous, N-th Order, Constant Coefficients","Duration":"6m 38s","ChapterTopicVideoID":7755,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this clip, I\u0027ll be talking about differential equations that are linear,"},{"Start":"00:05.040 ","End":"00:07.890","Text":"homogeneous, and have constant coefficients."},{"Start":"00:07.890 ","End":"00:10.215","Text":"Homogeneous means the 0 here,"},{"Start":"00:10.215 ","End":"00:16.215","Text":"the constant coefficients means that this as are numbers, not functions."},{"Start":"00:16.215 ","End":"00:19.890","Text":"This is going to be a bit similar to the"},{"Start":"00:19.890 ","End":"00:23.790","Text":"second order linear homogeneous with constant coefficients,"},{"Start":"00:23.790 ","End":"00:26.280","Text":"but we\u0027re taking in general nth order,"},{"Start":"00:26.280 ","End":"00:29.115","Text":"n I presume is at least 2."},{"Start":"00:29.115 ","End":"00:36.465","Text":"I\u0027m also presuming assuming that the coefficients are real numbers, not complex."},{"Start":"00:36.465 ","End":"00:39.230","Text":"This will be of order n according to the highest order"},{"Start":"00:39.230 ","End":"00:44.330","Text":"derivative and I\u0027m going to assume that the leading coefficient\u0027s not 0."},{"Start":"00:44.330 ","End":"00:46.850","Text":"Otherwise, why I put it in and the order will be lower?"},{"Start":"00:46.850 ","End":"00:48.259","Text":"It doesn\u0027t say here,"},{"Start":"00:48.259 ","End":"00:52.490","Text":"but usually assume the independent variable is x."},{"Start":"00:52.490 ","End":"00:57.290","Text":"We want to solve this for y as a function of x."},{"Start":"00:57.290 ","End":"00:59.180","Text":"We\u0027ll break it up into steps and again,"},{"Start":"00:59.180 ","End":"01:04.775","Text":"I want to suggest you go over the Order 2 theory because there are a lot of similarities."},{"Start":"01:04.775 ","End":"01:06.200","Text":"Anyway, just like there,"},{"Start":"01:06.200 ","End":"01:09.000","Text":"we have a characteristic equation here and"},{"Start":"01:09.000 ","End":"01:12.230","Text":"this characteristic equation is something that we have to solve."},{"Start":"01:12.230 ","End":"01:18.625","Text":"The way we get it is just the coefficients are copied and whatever order derivative,"},{"Start":"01:18.625 ","End":"01:22.185","Text":"let\u0027s say, y double prime is a second-order, we put a k^2."},{"Start":"01:22.185 ","End":"01:27.230","Text":"Nth order, we put k to the n. If it\u0027s just y itself,"},{"Start":"01:27.230 ","End":"01:31.535","Text":"not derived, then we just leave this as a constant times 1."},{"Start":"01:31.535 ","End":"01:33.980","Text":"We get this equation,"},{"Start":"01:33.980 ","End":"01:36.420","Text":"which is a polynomial equation."},{"Start":"01:36.420 ","End":"01:42.035","Text":"In general, a polynomial of degree n will have n solutions."},{"Start":"01:42.035 ","End":"01:43.595","Text":"Let\u0027s label them k_1,"},{"Start":"01:43.595 ","End":"01:50.389","Text":"k_2 up to k_n and when I say that there\u0027s n roots or n solutions to this equation,"},{"Start":"01:50.389 ","End":"01:53.735","Text":"that assumes that we\u0027re allowing for multiple roots."},{"Start":"01:53.735 ","End":"01:56.210","Text":"A root could repeat itself and it also"},{"Start":"01:56.210 ","End":"01:59.465","Text":"assumes that we\u0027re in the realm of complex numbers."},{"Start":"01:59.465 ","End":"02:01.346","Text":"Otherwise, it wouldn\u0027t be true."},{"Start":"02:01.346 ","End":"02:06.665","Text":"Some equations like this don\u0027t have any solutions at all in real numbers."},{"Start":"02:06.665 ","End":"02:09.905","Text":"We\u0027re allowing the multiple and complex."},{"Start":"02:09.905 ","End":"02:11.750","Text":"That\u0027s Step 1."},{"Start":"02:11.750 ","End":"02:12.850","Text":"In the second step,"},{"Start":"02:12.850 ","End":"02:16.070","Text":"we take these n roots and we sort and classify them"},{"Start":"02:16.070 ","End":"02:19.910","Text":"into groups and possibly even subgroups"},{"Start":"02:19.910 ","End":"02:23.240","Text":"and each such group or subgroup"},{"Start":"02:23.240 ","End":"02:27.620","Text":"provides a contribution to the general solution of a differential equation."},{"Start":"02:27.620 ","End":"02:29.240","Text":"When I say contribution,"},{"Start":"02:29.240 ","End":"02:32.600","Text":"each 1 of these provides a bit and then we add all the bits"},{"Start":"02:32.600 ","End":"02:36.470","Text":"together and I\u0027ll proceed and you\u0027ll see what I mean."},{"Start":"02:36.470 ","End":"02:38.330","Text":"Let\u0027s get some more space here."},{"Start":"02:38.330 ","End":"02:40.090","Text":"Let\u0027s look at the first group."},{"Start":"02:40.090 ","End":"02:42.770","Text":"Here, we take all the single,"},{"Start":"02:42.770 ","End":"02:46.535","Text":"not duplicated, not repeated, real roots."},{"Start":"02:46.535 ","End":"02:48.170","Text":"There might not be any."},{"Start":"02:48.170 ","End":"02:52.820","Text":"They could be anywhere from none up to n of them."},{"Start":"02:52.820 ","End":"02:56.870","Text":"Let\u0027s say we have m of them and if it\u0027s not an empty set,"},{"Start":"02:56.870 ","End":"02:59.480","Text":"then each 1 provides a contribution."},{"Start":"02:59.480 ","End":"03:02.390","Text":"Well, together they provide this here,"},{"Start":"03:02.390 ","End":"03:04.340","Text":"but it\u0027s really made up of separate bits."},{"Start":"03:04.340 ","End":"03:06.035","Text":"For example, the k_2,"},{"Start":"03:06.035 ","End":"03:10.700","Text":"we take e^k_2 times x and put a constant in front of it."},{"Start":"03:10.700 ","End":"03:12.050","Text":"For each of these,"},{"Start":"03:12.050 ","End":"03:14.510","Text":"each of the power of whatever it is times"},{"Start":"03:14.510 ","End":"03:17.540","Text":"x and a constant in front of it, different constants."},{"Start":"03:17.540 ","End":"03:22.175","Text":"That\u0027s for all the different lone wolf real roots."},{"Start":"03:22.175 ","End":"03:25.650","Text":"Now, the next group is also going to be"},{"Start":"03:25.650 ","End":"03:30.190","Text":"real and here we take the 1s that are multiple or repeated,"},{"Start":"03:30.190 ","End":"03:32.997","Text":"but there could be several like subgroups."},{"Start":"03:32.997 ","End":"03:36.410","Text":"For example, I could have 3 7s and then I could have 2 5s"},{"Start":"03:36.410 ","End":"03:40.205","Text":"and maybe I have 4 8s or whatever."},{"Start":"03:40.205 ","End":"03:42.050","Text":"I group them according to k,"},{"Start":"03:42.050 ","End":"03:44.120","Text":"they\u0027ll be several with all the same k,"},{"Start":"03:44.120 ","End":"03:47.475","Text":"but they might be several subgroups like this."},{"Start":"03:47.475 ","End":"03:50.000","Text":"Each such repeated value, let\u0027s call it k,"},{"Start":"03:50.000 ","End":"03:52.730","Text":"if all these ks are the same and just use the same letter,"},{"Start":"03:52.730 ","End":"03:54.502","Text":"which is going to be more than 1,"},{"Start":"03:54.502 ","End":"03:58.800","Text":"otherwise it wouldn\u0027t be repeated, then it contributes."},{"Start":"03:58.800 ","End":"04:00.467","Text":"It\u0027s a bit similar to this."},{"Start":"04:00.467 ","End":"04:03.710","Text":"We start off with e^kx with a constant."},{"Start":"04:03.710 ","End":"04:06.830","Text":"But instead of varying this k because they\u0027re all the same,"},{"Start":"04:06.830 ","End":"04:11.135","Text":"we just keep tacking an x onto the product."},{"Start":"04:11.135 ","End":"04:13.935","Text":"Here we just have it as it is here x, here x^2,"},{"Start":"04:13.935 ","End":"04:18.160","Text":"and so on up to x^p minus 1,"},{"Start":"04:18.160 ","End":"04:20.025","Text":"if there are p of them."},{"Start":"04:20.025 ","End":"04:21.630","Text":"Of course, if we have another k,"},{"Start":"04:21.630 ","End":"04:22.945","Text":"it may be an l,"},{"Start":"04:22.945 ","End":"04:26.030","Text":"which also with several repeated values,"},{"Start":"04:26.030 ","End":"04:27.718","Text":"then we have a similar thing to this,"},{"Start":"04:27.718 ","End":"04:30.555","Text":"but we can\u0027t reuse the c_1, c_2."},{"Start":"04:30.555 ","End":"04:32.760","Text":"We just continue the counting of the c,"},{"Start":"04:32.760 ","End":"04:35.060","Text":"but we\u0027ll see this in some of the examples."},{"Start":"04:35.060 ","End":"04:37.805","Text":"The third group are the complex roots."},{"Start":"04:37.805 ","End":"04:40.385","Text":"Now, they don\u0027t come on their own,"},{"Start":"04:40.385 ","End":"04:41.965","Text":"they come in pairs."},{"Start":"04:41.965 ","End":"04:45.320","Text":"It\u0027s maybe a theorem less than a theorem,"},{"Start":"04:45.320 ","End":"04:49.835","Text":"but that if we have 1 of those differential equations with real coefficients,"},{"Start":"04:49.835 ","End":"04:51.890","Text":"if I have a complex root,"},{"Start":"04:51.890 ","End":"04:53.730","Text":"its conjugate is also a root,"},{"Start":"04:53.730 ","End":"04:57.440","Text":"so they come in pairs like a plus or minus bi."},{"Start":"04:57.440 ","End":"04:59.720","Text":"That will be 2 different ks,"},{"Start":"04:59.720 ","End":"05:03.185","Text":"k_1 with the plus, k_2 with the minus, or whatever."},{"Start":"05:03.185 ","End":"05:09.020","Text":"But it gets a bit more complicated than that because even though they come in pairs,"},{"Start":"05:09.020 ","End":"05:10.759","Text":"the pairs could be distinct,"},{"Start":"05:10.759 ","End":"05:12.200","Text":"like in the first group,"},{"Start":"05:12.200 ","End":"05:13.400","Text":"or there could be multiple,"},{"Start":"05:13.400 ","End":"05:18.515","Text":"like in the second group only this time we\u0027re in complex roots organized into pairs."},{"Start":"05:18.515 ","End":"05:22.970","Text":"Here I put in words what I just said that the pairs could be distinct,"},{"Start":"05:22.970 ","End":"05:28.320","Text":"each a plus or minus bi on its own or I could have repetition, say,"},{"Start":"05:28.320 ","End":"05:30.990","Text":"3 times a plus or minus bi,"},{"Start":"05:30.990 ","End":"05:33.345","Text":"that would be maybe k_1 and 2,"},{"Start":"05:33.345 ","End":"05:35.715","Text":"k_3 and 4, k_5 and 6."},{"Start":"05:35.715 ","End":"05:38.400","Text":"I have to tell you what these contribute."},{"Start":"05:38.400 ","End":"05:42.440","Text":"I\u0027ve separated the distinct ones from the repeating multiple ones."},{"Start":"05:42.440 ","End":"05:50.011","Text":"The distinct ones, each distinct a plus or minus bi gives us this expression."},{"Start":"05:50.011 ","End":"05:51.463","Text":"As in all of these,"},{"Start":"05:51.463 ","End":"05:56.098","Text":"the indexes might have to be adjusted because all the indexes are going to be different,"},{"Start":"05:56.098 ","End":"06:00.095","Text":"but I always start from 1 again here just in the explanation."},{"Start":"06:00.095 ","End":"06:03.200","Text":"That\u0027s the expression. Now, if it repeats,"},{"Start":"06:03.200 ","End":"06:05.165","Text":"like say there\u0027s 3 of them,"},{"Start":"06:05.165 ","End":"06:08.285","Text":"then I do something similar like I did in the second group."},{"Start":"06:08.285 ","End":"06:11.390","Text":"I make repetitions of this thing with"},{"Start":"06:11.390 ","End":"06:16.100","Text":"the difference that I keep sticking an extra x on each time."},{"Start":"06:16.100 ","End":"06:18.770","Text":"First of all as is, then x, then x^2,"},{"Start":"06:18.770 ","End":"06:24.000","Text":"I just took the example where there were triple group and the constants just differ."},{"Start":"06:24.000 ","End":"06:26.070","Text":"They\u0027re just running index."},{"Start":"06:26.070 ","End":"06:30.920","Text":"Here we\u0027d get 6 different constants because each pair is 2 and then it\u0027s a triple."},{"Start":"06:30.920 ","End":"06:34.340","Text":"Anyway, all this will be clearer after you\u0027ve seen a"},{"Start":"06:34.340 ","End":"06:38.640","Text":"few of the solved examples and I\u0027m going to stop here."}],"ID":7812},{"Watched":false,"Name":"Useful Theorems on Polynomial Roots","Duration":"9m 5s","ChapterTopicVideoID":7756,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.000","Text":"The purpose of this clip is to bring you some useful results,"},{"Start":"00:04.000 ","End":"00:07.410","Text":"or theorems for solving the characteristic equation,"},{"Start":"00:07.410 ","End":"00:11.999","Text":"which is a polynomial equation of degree n in general,"},{"Start":"00:11.999 ","End":"00:15.540","Text":"and collected together a bunch of useful stuff."},{"Start":"00:15.540 ","End":"00:17.340","Text":"Let\u0027s start with the first one."},{"Start":"00:17.340 ","End":"00:18.540","Text":"You\u0027ve probably said it before,"},{"Start":"00:18.540 ","End":"00:21.495","Text":"but when we have a polynomial equation of degree n,"},{"Start":"00:21.495 ","End":"00:23.925","Text":"then it\u0027s going to have exactly n roots,"},{"Start":"00:23.925 ","End":"00:30.435","Text":"provided that we count multiple roots as separate count."},{"Start":"00:30.435 ","End":"00:32.490","Text":"I mean, if something repeats 3 times,"},{"Start":"00:32.490 ","End":"00:33.870","Text":"we give it a count of 3,"},{"Start":"00:33.870 ","End":"00:36.240","Text":"and also we have to include complex numbers,"},{"Start":"00:36.240 ","End":"00:38.100","Text":"not just the real roots."},{"Start":"00:38.100 ","End":"00:41.400","Text":"Then degree n, exactly n roots."},{"Start":"00:41.400 ","End":"00:42.875","Text":"I\u0027ll give an example."},{"Start":"00:42.875 ","End":"00:46.475","Text":"Here we have a degree 4 polynomial equation."},{"Start":"00:46.475 ","End":"00:50.255","Text":"We often use the letter z when we work with complex numbers."},{"Start":"00:50.255 ","End":"00:52.940","Text":"Degree is 4 and it has 4 roots."},{"Start":"00:52.940 ","End":"00:57.710","Text":"Basically, z^4 is 1 and we need the 4 4th roots of 1,"},{"Start":"00:57.710 ","End":"00:59.620","Text":"and there are 1, i,"},{"Start":"00:59.620 ","End":"01:01.675","Text":"minus i, minus 1."},{"Start":"01:01.675 ","End":"01:06.440","Text":"Another example, this polynomial in x of degree 3,"},{"Start":"01:06.440 ","End":"01:08.990","Text":"it actually has 3 roots and they\u0027re all the same."},{"Start":"01:08.990 ","End":"01:14.825","Text":"They\u0027re all equal to 1 because this one is actually x minus 1^3."},{"Start":"01:14.825 ","End":"01:18.140","Text":"I just wanted to include some multiple roots,"},{"Start":"01:18.140 ","End":"01:21.155","Text":"and complex roots just so you remember that."},{"Start":"01:21.155 ","End":"01:23.180","Text":"Degree 4 has 4 roots,"},{"Start":"01:23.180 ","End":"01:24.350","Text":"degree 3 has 3 roots,"},{"Start":"01:24.350 ","End":"01:26.645","Text":"and in general degree n, n roots."},{"Start":"01:26.645 ","End":"01:28.985","Text":"Let\u0027s go onto the next, to number 2."},{"Start":"01:28.985 ","End":"01:32.615","Text":"I just want to remind you that we\u0027re dealing with polynomials with real coefficients."},{"Start":"01:32.615 ","End":"01:35.150","Text":"Otherwise, number 2 wouldn\u0027t apply."},{"Start":"01:35.150 ","End":"01:37.840","Text":"But if we find 1 complex root,"},{"Start":"01:37.840 ","End":"01:40.075","Text":"a plus bi,"},{"Start":"01:40.075 ","End":"01:44.165","Text":"then its conjugate a minus bi will also be a root."},{"Start":"01:44.165 ","End":"01:46.070","Text":"In other words, the complex roots,"},{"Start":"01:46.070 ","End":"01:47.355","Text":"they travel in pairs,"},{"Start":"01:47.355 ","End":"01:49.490","Text":"a plus, or minus bi."},{"Start":"01:49.490 ","End":"01:51.205","Text":"In number 3,"},{"Start":"01:51.205 ","End":"01:54.560","Text":"I\u0027m going to have a polynomial with integer coefficients,"},{"Start":"01:54.560 ","End":"01:57.770","Text":"and also let\u0027s make the leading coefficient 1."},{"Start":"01:57.770 ","End":"01:59.990","Text":"If it isn\u0027t 1, we can always divide by it."},{"Start":"01:59.990 ","End":"02:03.160","Text":"Well, let\u0027s assume that we still have integer coefficients."},{"Start":"02:03.160 ","End":"02:05.360","Text":"Then in that case,"},{"Start":"02:05.360 ","End":"02:08.690","Text":"if I want to search for integer roots,"},{"Start":"02:08.690 ","End":"02:12.139","Text":"so solutions for p(x) equals 0,"},{"Start":"02:12.139 ","End":"02:19.700","Text":"then I am very restricted in that it has to be a divisor of this constant term a_0,"},{"Start":"02:19.700 ","End":"02:22.475","Text":"and I brought a good example here."},{"Start":"02:22.475 ","End":"02:25.790","Text":"We\u0027re looking for integer solutions for"},{"Start":"02:25.790 ","End":"02:29.555","Text":"this equation or integer roots for this polynomial, same thing."},{"Start":"02:29.555 ","End":"02:33.590","Text":"Notice that it starts with a 1 or we don\u0027t see the 1, but anyway."},{"Start":"02:33.590 ","End":"02:35.750","Text":"If we want the integer solutions,"},{"Start":"02:35.750 ","End":"02:37.760","Text":"they have to divide the minus 6,"},{"Start":"02:37.760 ","End":"02:40.115","Text":"and we take both the plus, and the minus,"},{"Start":"02:40.115 ","End":"02:43.670","Text":"since the divisors of 6 are 1, 2, 3, and 6,"},{"Start":"02:43.670 ","End":"02:46.730","Text":"we\u0027re altogether actually got 8 possibilities to check;"},{"Start":"02:46.730 ","End":"02:47.915","Text":"plus or minus 1,"},{"Start":"02:47.915 ","End":"02:49.655","Text":"up to plus or minus 6."},{"Start":"02:49.655 ","End":"02:50.840","Text":"Just take each one,"},{"Start":"02:50.840 ","End":"02:53.780","Text":"substitute, and see if you get 0 or not."},{"Start":"02:53.780 ","End":"02:55.190","Text":"For example, put in 1,"},{"Start":"02:55.190 ","End":"02:58.355","Text":"get 1 minus 6 plus 11 minus 6,"},{"Start":"02:58.355 ","End":"03:00.095","Text":"12 minus 12, 0."},{"Start":"03:00.095 ","End":"03:01.760","Text":"Turns out that of all these 8,"},{"Start":"03:01.760 ","End":"03:05.345","Text":"just these 3 are integer roots."},{"Start":"03:05.345 ","End":"03:07.760","Text":"Because I\u0027ve already found 3 of them in actual fact,"},{"Start":"03:07.760 ","End":"03:09.050","Text":"I know that these are all the roots,"},{"Start":"03:09.050 ","End":"03:10.280","Text":"there can\u0027t be anymore."},{"Start":"03:10.280 ","End":"03:13.150","Text":"Let\u0027s move onto number 4."},{"Start":"03:13.150 ","End":"03:17.135","Text":"But, I\u0027ll give myself some more space first. Let\u0027s see."},{"Start":"03:17.135 ","End":"03:21.775","Text":"This one says that if I have a polynomial and if I find 1 root,"},{"Start":"03:21.775 ","End":"03:23.595","Text":"say x equals a,"},{"Start":"03:23.595 ","End":"03:25.080","Text":"of this polynomial,"},{"Start":"03:25.080 ","End":"03:31.265","Text":"then this polynomial is evenly divisible by x minus a, no remainder."},{"Start":"03:31.265 ","End":"03:35.435","Text":"This is useful for reducing the degree of the polynomial."},{"Start":"03:35.435 ","End":"03:41.030","Text":"Especially say, if we have a cubic and you find 1 solution a,"},{"Start":"03:41.030 ","End":"03:44.180","Text":"then you divide by x minus a and you get a quadratic,"},{"Start":"03:44.180 ","End":"03:46.310","Text":"and a quadratic, we already know how to solve them;"},{"Start":"03:46.310 ","End":"03:47.900","Text":"we get the other 2 roots."},{"Start":"03:47.900 ","End":"03:49.460","Text":"I\u0027m not going to give an example."},{"Start":"03:49.460 ","End":"03:51.110","Text":"Let\u0027s move on to number 5."},{"Start":"03:51.110 ","End":"03:58.010","Text":"In this one, I have a polynomial p(x) and I have that a is a root of p,"},{"Start":"03:58.010 ","End":"04:01.190","Text":"but also a root of the derivative."},{"Start":"04:01.190 ","End":"04:06.110","Text":"In other words, p(a) is 0 and p\u0027(a) is also 0."},{"Start":"04:06.110 ","End":"04:07.430","Text":"If this happens,"},{"Start":"04:07.430 ","End":"04:10.310","Text":"then it\u0027s guaranteed that a is a double root."},{"Start":"04:10.310 ","End":"04:11.420","Text":"When I say double root,"},{"Start":"04:11.420 ","End":"04:13.250","Text":"I mean at least double root."},{"Start":"04:13.250 ","End":"04:15.860","Text":"Like in real life, if 2 guys are twins,"},{"Start":"04:15.860 ","End":"04:17.870","Text":"they might not be twins, they might be triplets,"},{"Start":"04:17.870 ","End":"04:19.940","Text":"but they\u0027re still twins anyway."},{"Start":"04:19.940 ","End":"04:23.585","Text":"Double or more, it could be triple, or quadruple."},{"Start":"04:23.585 ","End":"04:26.120","Text":"Now, each time I differentiate and get 0,"},{"Start":"04:26.120 ","End":"04:29.975","Text":"so the second derivative also is 0 when x is a,"},{"Start":"04:29.975 ","End":"04:33.020","Text":"so a is a root of p\u0027\u0027,"},{"Start":"04:33.020 ","End":"04:36.340","Text":"then I know that a is a triple root and so on."},{"Start":"04:36.340 ","End":"04:39.570","Text":"If also p\u0027\u0027\u0027(a) is 0,"},{"Start":"04:39.570 ","End":"04:42.565","Text":"it\u0027s a quadruple root, etc."},{"Start":"04:42.565 ","End":"04:44.309","Text":"I\u0027m going to give an example."},{"Start":"04:44.309 ","End":"04:46.630","Text":"Let\u0027s take example,"},{"Start":"04:46.630 ","End":"04:49.690","Text":"I think actually this should have been an Example 4."},{"Start":"04:49.690 ","End":"04:51.910","Text":"Not that it really matters, I lost count."},{"Start":"04:51.910 ","End":"04:53.025","Text":"This is Example 4."},{"Start":"04:53.025 ","End":"04:57.245","Text":"If we take this polynomial which I borrowed from Example 2 above,"},{"Start":"04:57.245 ","End":"05:01.625","Text":"then notice that if I differentiate it,"},{"Start":"05:01.625 ","End":"05:05.040","Text":"I get this, and the second derivative is this."},{"Start":"05:05.040 ","End":"05:07.939","Text":"I have a typo here."},{"Start":"05:07.939 ","End":"05:09.875","Text":"Instead of a, it should be 1."},{"Start":"05:09.875 ","End":"05:11.570","Text":"This is 1,"},{"Start":"05:11.570 ","End":"05:13.655","Text":"1, and 1; sorry about that."},{"Start":"05:13.655 ","End":"05:19.205","Text":"Then p(1) is 0 because look 1 minus 3 plus 3 minus 1 is 0."},{"Start":"05:19.205 ","End":"05:25.634","Text":"P\u0027(1), I get 3 minus 6 plus 3 is also 0."},{"Start":"05:25.634 ","End":"05:27.030","Text":"If I put 1 in here,"},{"Start":"05:27.030 ","End":"05:29.640","Text":"I get 6 minus 6, which is 0."},{"Start":"05:29.640 ","End":"05:33.493","Text":"It\u0027s a root of the polynomials derivative,"},{"Start":"05:33.493 ","End":"05:34.730","Text":"and a second derivative,"},{"Start":"05:34.730 ","End":"05:37.670","Text":"and that means that 1 is a triple root,"},{"Start":"05:37.670 ","End":"05:42.335","Text":"and this is actually what we found also in Example 2,"},{"Start":"05:42.335 ","End":"05:48.830","Text":"and I mentioned that this is actually x minus 1^3,"},{"Start":"05:48.830 ","End":"05:52.175","Text":"which is the reason why this happens."},{"Start":"05:52.175 ","End":"05:56.515","Text":"That\u0027s the 5 and 1 more to go."},{"Start":"05:56.515 ","End":"05:59.270","Text":"In number 6, I want to talk a bit about nth roots,"},{"Start":"05:59.270 ","End":"06:01.970","Text":"which is treated slightly differently or quite"},{"Start":"06:01.970 ","End":"06:05.435","Text":"differently in complex numbers as opposed to real numbers."},{"Start":"06:05.435 ","End":"06:10.160","Text":"Now, if you have real numbers and if I have an equation x squared minus 9,"},{"Start":"06:10.160 ","End":"06:11.390","Text":"it has 2 solutions;"},{"Start":"06:11.390 ","End":"06:13.020","Text":"3 and minus 3,"},{"Start":"06:13.020 ","End":"06:15.230","Text":"or sometimes written plus, or minus 3."},{"Start":"06:15.230 ","End":"06:21.230","Text":"Some people would like to say that the square root of 9 is 3 or minus 3,"},{"Start":"06:21.230 ","End":"06:25.130","Text":"but then that wouldn\u0027t be uniquely defined as a function."},{"Start":"06:25.130 ","End":"06:28.145","Text":"We usually write that the square root of 9 is just 3."},{"Start":"06:28.145 ","End":"06:34.370","Text":"But when loosely speaking you could say that 3 and minus 3 are both square roots of 9."},{"Start":"06:34.370 ","End":"06:36.710","Text":"At any rate, it\u0027s true that both of them satisfy"},{"Start":"06:36.710 ","End":"06:39.620","Text":"this equation that when you square them you get 9."},{"Start":"06:39.620 ","End":"06:41.240","Text":"So much for real numbers."},{"Start":"06:41.240 ","End":"06:43.600","Text":"Now, what about complex numbers?"},{"Start":"06:43.600 ","End":"06:45.780","Text":"If you go back and look at Example 1,"},{"Start":"06:45.780 ","End":"06:49.400","Text":"where we had x^4 is 1 or what was it?"},{"Start":"06:49.400 ","End":"06:53.044","Text":"I think it was z^4 minus 1 equals 0."},{"Start":"06:53.044 ","End":"06:54.860","Text":"We got 4 solutions,"},{"Start":"06:54.860 ","End":"06:56.450","Text":"1, i minus 1,"},{"Start":"06:56.450 ","End":"06:57.620","Text":"and minus i,"},{"Start":"06:57.620 ","End":"07:00.395","Text":"and we expect 4 solutions to the fourth degree."},{"Start":"07:00.395 ","End":"07:04.040","Text":"In complex numbers, you more often loosely say"},{"Start":"07:04.040 ","End":"07:07.640","Text":"that the fourth root of 1 is any one of these."},{"Start":"07:07.640 ","End":"07:12.110","Text":"There\u0027s no one that stands out as a favorite like with real numbers,"},{"Start":"07:12.110 ","End":"07:14.710","Text":"the positive 1 is the 1."},{"Start":"07:14.710 ","End":"07:19.280","Text":"Sometimes you say that any one of these is the fourth root of 1."},{"Start":"07:19.280 ","End":"07:23.725","Text":"Anyway, they both satisfy the equation that to the power of 4 is 1."},{"Start":"07:23.725 ","End":"07:26.495","Text":"In this situation, that\u0027s something that occurs a lot."},{"Start":"07:26.495 ","End":"07:31.340","Text":"We\u0027re looking for whole n solutions of z^n equals"},{"Start":"07:31.340 ","End":"07:38.185","Text":"some constant given complex number z_0 in this case 1."},{"Start":"07:38.185 ","End":"07:43.235","Text":"Formally, we want all the nth roots of z_0."},{"Start":"07:43.235 ","End":"07:45.650","Text":"The formula which is here in blue,"},{"Start":"07:45.650 ","End":"07:47.240","Text":"I\u0027ll get to it in a moment,"},{"Start":"07:47.240 ","End":"07:52.115","Text":"is based on the z_0 being in polar form."},{"Start":"07:52.115 ","End":"07:57.575","Text":"You can go back and review that when we have a complex number instead of a plus bi,"},{"Start":"07:57.575 ","End":"08:00.890","Text":"we break it up into a modulus in an argument."},{"Start":"08:00.890 ","End":"08:02.450","Text":"The modulus is r,"},{"Start":"08:02.450 ","End":"08:04.415","Text":"which is the distance from the origin,"},{"Start":"08:04.415 ","End":"08:05.770","Text":"and Theta is the angle."},{"Start":"08:05.770 ","End":"08:09.355","Text":"It\u0027s a bit like converting Cartesian to polar."},{"Start":"08:09.355 ","End":"08:12.140","Text":"When we have z_0 in this form,"},{"Start":"08:12.140 ","End":"08:14.060","Text":"then the nth roots,"},{"Start":"08:14.060 ","End":"08:17.330","Text":"or should I say the n solutions of this equation,"},{"Start":"08:17.330 ","End":"08:19.880","Text":"z_1 through z_n,"},{"Start":"08:19.880 ","End":"08:21.500","Text":"given by this formula,"},{"Start":"08:21.500 ","End":"08:24.845","Text":"we take the positive nth root of"},{"Start":"08:24.845 ","End":"08:30.440","Text":"the positive number a and then we multiply it by n different things."},{"Start":"08:30.440 ","End":"08:36.650","Text":"K is an index that runs from 0 to n minus 1 because there\u0027s n different ones."},{"Start":"08:36.650 ","End":"08:38.884","Text":"We just plug into this formula,"},{"Start":"08:38.884 ","End":"08:43.280","Text":"different values of k and this gives us,"},{"Start":"08:43.280 ","End":"08:45.770","Text":"also in polar form,"},{"Start":"08:45.770 ","End":"08:52.250","Text":"the n solutions or if you like the nth roots of z_0."},{"Start":"08:52.250 ","End":"08:55.550","Text":"You could actually continue with k beyond n minus 1,"},{"Start":"08:55.550 ","End":"08:57.995","Text":"but then they start repeating themselves."},{"Start":"08:57.995 ","End":"08:59.900","Text":"To get them all distinct,"},{"Start":"08:59.900 ","End":"09:03.035","Text":"you just take these values of k,"},{"Start":"09:03.035 ","End":"09:06.240","Text":"and I\u0027m not going to give an example."}],"ID":7813},{"Watched":false,"Name":"Exercise 1","Duration":"56s","ChapterTopicVideoID":7739,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"We have this differential equation to solve,"},{"Start":"00:02.370 ","End":"00:04.620","Text":"linear homogeneous constant coefficient."},{"Start":"00:04.620 ","End":"00:09.180","Text":"So we start with the characteristic equation which we get, as you know,"},{"Start":"00:09.180 ","End":"00:10.950","Text":"triple prime is k^3,"},{"Start":"00:10.950 ","End":"00:13.110","Text":"y\" we replace with k^2,"},{"Start":"00:13.110 ","End":"00:15.420","Text":"y\u0027 we replace with k and so on."},{"Start":"00:15.420 ","End":"00:18.330","Text":"Now it\u0027s a cubic equation, but luckily,"},{"Start":"00:18.330 ","End":"00:21.900","Text":"we can take k outside the brackets and this is what we get."},{"Start":"00:21.900 ","End":"00:26.550","Text":"So either k is 0 or k is one of the two solutions of this quadratic."},{"Start":"00:26.550 ","End":"00:29.130","Text":"In short, these are the three solutions,"},{"Start":"00:29.130 ","End":"00:32.250","Text":"0, minus 1, and 3 and notice that they\u0027re all different."},{"Start":"00:32.250 ","End":"00:35.240","Text":"So now we can go on to the next step of writing"},{"Start":"00:35.240 ","End":"00:39.710","Text":"the general solution to the differential equation and this is what we get."},{"Start":"00:39.710 ","End":"00:41.449","Text":"In each piece is a constant,"},{"Start":"00:41.449 ","End":"00:45.625","Text":"e to the power of whatever it is here times x,"},{"Start":"00:45.625 ","End":"00:47.760","Text":"0, minus 1, and 3."},{"Start":"00:47.760 ","End":"00:56.790","Text":"However, e^0x is 1 so we can simplify and write our solution like this and we\u0027re done."}],"ID":7814},{"Watched":false,"Name":"Exercise 2","Duration":"1m 38s","ChapterTopicVideoID":7740,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"We have this differential equation to solve,"},{"Start":"00:02.280 ","End":"00:04.995","Text":"it\u0027s linear homogeneous constant coefficient,"},{"Start":"00:04.995 ","End":"00:08.160","Text":"so we go for the characteristic equation first."},{"Start":"00:08.160 ","End":"00:10.965","Text":"Here it is, as you know,"},{"Start":"00:10.965 ","End":"00:13.560","Text":"fourth derivative is k^4,"},{"Start":"00:13.560 ","End":"00:16.155","Text":"third derivative k^3 and so on."},{"Start":"00:16.155 ","End":"00:17.340","Text":"Now how do we solve this?"},{"Start":"00:17.340 ","End":"00:19.860","Text":"We don\u0027t know how to solve fourth degree equations,"},{"Start":"00:19.860 ","End":"00:23.985","Text":"but there is a theorem that when the leading coefficient is 1, which it is,"},{"Start":"00:23.985 ","End":"00:30.360","Text":"then any route or solution to this has to divide the last coefficient,"},{"Start":"00:30.360 ","End":"00:33.115","Text":"the 30, could be plus or minus."},{"Start":"00:33.115 ","End":"00:36.080","Text":"Now, 30 certainly has a lot of divisors, 1, 2,"},{"Start":"00:36.080 ","End":"00:39.275","Text":"5, 6, 10, 15 and 30 and each could be plus or minus."},{"Start":"00:39.275 ","End":"00:40.520","Text":"There\u0027s quite a few. How many?"},{"Start":"00:40.520 ","End":"00:42.340","Text":"There\u0027s 14 numbers to check."},{"Start":"00:42.340 ","End":"00:44.605","Text":"We need 4 solutions."},{"Start":"00:44.605 ","End":"00:46.365","Text":"We just try them one-by-one."},{"Start":"00:46.365 ","End":"00:49.505","Text":"For example, if you let k=1,"},{"Start":"00:49.505 ","End":"00:56.905","Text":"then what you get is, 1+3-15-19+30."},{"Start":"00:56.905 ","End":"00:58.820","Text":"That works out okay, because look,"},{"Start":"00:58.820 ","End":"01:01.535","Text":"plus is 34 minus 34,"},{"Start":"01:01.535 ","End":"01:05.825","Text":"so it is 0, so we have k=1 and so on."},{"Start":"01:05.825 ","End":"01:10.850","Text":"Keep trying and you\u0027ll find that there are 4 solutions,"},{"Start":"01:10.850 ","End":"01:12.080","Text":"and here they are."},{"Start":"01:12.080 ","End":"01:14.930","Text":"You can check if you like, substitute each one."},{"Start":"01:14.930 ","End":"01:17.105","Text":"Once we get the 4 solutions, we stop."},{"Start":"01:17.105 ","End":"01:20.040","Text":"Here, all 4 of them are different."},{"Start":"01:20.040 ","End":"01:21.470","Text":"When they\u0027re all different,"},{"Start":"01:21.470 ","End":"01:25.400","Text":"we now have to find the general solution of the differential equation, I mean."},{"Start":"01:25.400 ","End":"01:27.080","Text":"We just put different constants,"},{"Start":"01:27.080 ","End":"01:28.280","Text":"e to the power of,"},{"Start":"01:28.280 ","End":"01:32.195","Text":"and then each 1 times x. I didn\u0027t write 1x,"},{"Start":"01:32.195 ","End":"01:34.430","Text":"I wrote x, but there\u0027s the minus 2,"},{"Start":"01:34.430 ","End":"01:36.530","Text":"there\u0027s 3, and there\u0027s the minus 5."},{"Start":"01:36.530 ","End":"01:38.760","Text":"That\u0027s it. We\u0027re done."}],"ID":7815},{"Watched":false,"Name":"Exercise 3","Duration":"1m 21s","ChapterTopicVideoID":7741,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"We want to solve this differential equation,"},{"Start":"00:02.580 ","End":"00:05.865","Text":"linear homogeneous constant coefficients."},{"Start":"00:05.865 ","End":"00:09.840","Text":"We find the characteristic equation, which is this."},{"Start":"00:09.840 ","End":"00:11.880","Text":"It\u0027s a cubic equation,"},{"Start":"00:11.880 ","End":"00:14.010","Text":"so we either have to start guessing roots,"},{"Start":"00:14.010 ","End":"00:16.725","Text":"or we can try factorizing."},{"Start":"00:16.725 ","End":"00:20.790","Text":"We can use factorization in pairs here because we can see there\u0027s"},{"Start":"00:20.790 ","End":"00:25.140","Text":"a 1 and 2 with opposite signs and a 1 and a 2 with opposite signs."},{"Start":"00:25.140 ","End":"00:27.210","Text":"As you see, we take the k^2 out,"},{"Start":"00:27.210 ","End":"00:28.950","Text":"you got k minus 2."},{"Start":"00:28.950 ","End":"00:30.810","Text":"Then if we just take minus 1 out,"},{"Start":"00:30.810 ","End":"00:32.865","Text":"we\u0027ll also have k minus 2."},{"Start":"00:32.865 ","End":"00:35.340","Text":"Now k minus 2 is common to this,"},{"Start":"00:35.340 ","End":"00:37.620","Text":"so we can take it outside the brackets."},{"Start":"00:37.620 ","End":"00:39.870","Text":"What we\u0027re left with is k^2 from here,"},{"Start":"00:39.870 ","End":"00:41.940","Text":"minus 1 from here."},{"Start":"00:41.940 ","End":"00:44.930","Text":"This is now clear how to solve, because from here,"},{"Start":"00:44.930 ","End":"00:48.335","Text":"k is equal to 2, or k^2=1."},{"Start":"00:48.335 ","End":"00:50.240","Text":"Here I chose to factorize it,"},{"Start":"00:50.240 ","End":"00:54.710","Text":"so that we could\u0027ve taken a shortcut if k^2 is 1 and k is plus or minus 1,"},{"Start":"00:54.710 ","End":"00:55.925","Text":"maybe save the line."},{"Start":"00:55.925 ","End":"00:57.739","Text":"But this is more methodical."},{"Start":"00:57.739 ","End":"01:01.170","Text":"Here we see that the three solutions for k are 2,"},{"Start":"01:01.170 ","End":"01:02.670","Text":"1 and minus 1."},{"Start":"01:02.670 ","End":"01:04.010","Text":"All three are different."},{"Start":"01:04.010 ","End":"01:05.510","Text":"When we have all different solutions,"},{"Start":"01:05.510 ","End":"01:07.100","Text":"we know how to finish it off."},{"Start":"01:07.100 ","End":"01:11.315","Text":"For each root, like a constant e to the power of that number times x,"},{"Start":"01:11.315 ","End":"01:13.370","Text":"except here where I didn\u0027t write the 1x,"},{"Start":"01:13.370 ","End":"01:16.115","Text":"I just wrote it as x, just standard."},{"Start":"01:16.115 ","End":"01:18.305","Text":"Minus 1x is just minus x."},{"Start":"01:18.305 ","End":"01:21.690","Text":"Anyway, this is the solution and we\u0027re done."}],"ID":7816},{"Watched":false,"Name":"Exercise 4","Duration":"1m 42s","ChapterTopicVideoID":7742,"CourseChapterTopicPlaylistID":4228,"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 have this differential equation to solve of the familiar type."},{"Start":"00:04.770 ","End":"00:09.810","Text":"We know what we have to do first is to get the characteristic equation. Here it is."},{"Start":"00:09.810 ","End":"00:10.830","Text":"You know how to do this."},{"Start":"00:10.830 ","End":"00:14.190","Text":"Thing is, it\u0027s a fourth degree equation. What do we do?"},{"Start":"00:14.190 ","End":"00:17.115","Text":"Well, if you look closely, well, not so closely even,"},{"Start":"00:17.115 ","End":"00:18.720","Text":"you see that the odd powers are missing."},{"Start":"00:18.720 ","End":"00:20.520","Text":"There\u0027s no k^3 and there\u0027s no k,"},{"Start":"00:20.520 ","End":"00:25.590","Text":"so the standard trick is to substitute k^2 and let\u0027s make it t,"},{"Start":"00:25.590 ","End":"00:27.165","Text":"it\u0027s a popular letter,"},{"Start":"00:27.165 ","End":"00:28.410","Text":"and so if we do that,"},{"Start":"00:28.410 ","End":"00:30.780","Text":"we will get, this becomes t squared."},{"Start":"00:30.780 ","End":"00:33.495","Text":"This is just minus 5t plus 4 as here."},{"Start":"00:33.495 ","End":"00:36.860","Text":"Now it\u0027s a plain quadratic and this we know how to solve."},{"Start":"00:36.860 ","End":"00:38.555","Text":"I\u0027ll just give you the two solutions."},{"Start":"00:38.555 ","End":"00:44.280","Text":"We\u0027ve got that either t is 1 or t is 4."},{"Start":"00:44.280 ","End":"00:45.540","Text":"This is just t minus 1,"},{"Start":"00:45.540 ","End":"00:47.210","Text":"t minus 4 by factorization,"},{"Start":"00:47.210 ","End":"00:49.040","Text":"or use the formula or whatever,"},{"Start":"00:49.040 ","End":"00:50.495","Text":"the solutions are 1 and 4."},{"Start":"00:50.495 ","End":"00:51.860","Text":"But we\u0027re not looking for t,"},{"Start":"00:51.860 ","End":"00:54.450","Text":"we\u0027re looking for k, so we\u0027ll take them one at a time."},{"Start":"00:54.450 ","End":"00:55.890","Text":"If t is 1,"},{"Start":"00:55.890 ","End":"00:58.605","Text":"then t is k^2, so k^2 is 1."},{"Start":"00:58.605 ","End":"01:01.290","Text":"So k is plus or minus 1,"},{"Start":"01:01.290 ","End":"01:03.405","Text":"that\u0027s k_1 and k_2."},{"Start":"01:03.405 ","End":"01:04.980","Text":"Similarly, if t is 4,"},{"Start":"01:04.980 ","End":"01:06.270","Text":"k^2 is 4,"},{"Start":"01:06.270 ","End":"01:08.190","Text":"so k is plus or minus 2,"},{"Start":"01:08.190 ","End":"01:11.895","Text":"and that gives us k_3 and k_4."},{"Start":"01:11.895 ","End":"01:15.655","Text":"Notice that we have four different solutions for k. We have 1,"},{"Start":"01:15.655 ","End":"01:19.030","Text":"minus 1, 2, and minus 2."},{"Start":"01:19.030 ","End":"01:21.050","Text":"When we have all different solutions,"},{"Start":"01:21.050 ","End":"01:23.840","Text":"we know how to write the solution to the ODE."},{"Start":"01:23.840 ","End":"01:25.605","Text":"For example, for this two,"},{"Start":"01:25.605 ","End":"01:28.590","Text":"we put some constant e^2x."},{"Start":"01:28.590 ","End":"01:30.285","Text":"Here\u0027s the minus 2,"},{"Start":"01:30.285 ","End":"01:33.180","Text":"I don\u0027t write 1x and I don\u0027t write minus 1x,"},{"Start":"01:33.180 ","End":"01:34.890","Text":"the 1 stays out,"},{"Start":"01:34.890 ","End":"01:37.220","Text":"but anyway, this is the general solution."},{"Start":"01:37.220 ","End":"01:39.080","Text":"Of course, different constants, c_1,"},{"Start":"01:39.080 ","End":"01:42.390","Text":"c_2, c_3, c_4. Anyway, we\u0027re done."}],"ID":7817},{"Watched":false,"Name":"Exercise 5","Duration":"2m 2s","ChapterTopicVideoID":7743,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"Here we have this differential equation to solve."},{"Start":"00:02.790 ","End":"00:05.520","Text":"It\u0027s the kind we\u0027ve been solving in this chapter,"},{"Start":"00:05.520 ","End":"00:06.720","Text":"so we know what to do."},{"Start":"00:06.720 ","End":"00:09.630","Text":"We need the characteristic equation first,"},{"Start":"00:09.630 ","End":"00:13.230","Text":"and this is clearly k_4 minus 1 equals 0."},{"Start":"00:13.230 ","End":"00:15.780","Text":"It\u0027s 4th degree, but we can factorize it."},{"Start":"00:15.780 ","End":"00:21.720","Text":"Because k^4 is k^2^2 so we can use the difference of squares formula,"},{"Start":"00:21.720 ","End":"00:26.955","Text":"and then we factorize it as (k^2-1)(k^2-1)=0."},{"Start":"00:26.955 ","End":"00:28.770","Text":"As usual, if a product is 0,"},{"Start":"00:28.770 ","End":"00:31.830","Text":"then either one of them must be 0."},{"Start":"00:31.830 ","End":"00:35.820","Text":"Either k^2 minus 1 is 0 or k^2+1=0."},{"Start":"00:35.820 ","End":"00:36.975","Text":"Let\u0027s take them one at a time."},{"Start":"00:36.975 ","End":"00:40.155","Text":"The first one, k^2-1=0,"},{"Start":"00:40.155 ","End":"00:42.770","Text":"either factorize it or bring the 1 to the other side."},{"Start":"00:42.770 ","End":"00:46.025","Text":"Either way you get the 2 solutions are plus or minus 1."},{"Start":"00:46.025 ","End":"00:47.915","Text":"That\u0027s k_1 and k_2."},{"Start":"00:47.915 ","End":"00:51.430","Text":"Now, the next bit, k^2 plus 1 is 0."},{"Start":"00:51.430 ","End":"00:55.590","Text":"Easiest just to bring to the 1 to the other side, k^2 is minus 1."},{"Start":"00:55.590 ","End":"00:57.780","Text":"We\u0027re into complex numbers."},{"Start":"00:57.780 ","End":"01:03.090","Text":"The two solutions are the two square roots of minus 1 plus or minus i,"},{"Start":"01:03.090 ","End":"01:04.785","Text":"and that\u0027s k_3 and k_4,"},{"Start":"01:04.785 ","End":"01:06.025","Text":"i and minus i,"},{"Start":"01:06.025 ","End":"01:09.350","Text":"but rather write it as 0 plus or minus 1i."},{"Start":"01:09.350 ","End":"01:12.980","Text":"The reason I\u0027m doing that is because I don\u0027t want to use a formula which"},{"Start":"01:12.980 ","End":"01:16.445","Text":"says that when I have an a plus or minus bi,"},{"Start":"01:16.445 ","End":"01:20.060","Text":"then its contribution to the solution is this,"},{"Start":"01:20.060 ","End":"01:22.715","Text":"this is the a and this is the b here and here."},{"Start":"01:22.715 ","End":"01:24.515","Text":"However, besides this,"},{"Start":"01:24.515 ","End":"01:30.125","Text":"remember we also have from here the 1 and the minus 1, and they\u0027re different."},{"Start":"01:30.125 ","End":"01:32.165","Text":"This part, we know how to deal with."},{"Start":"01:32.165 ","End":"01:36.015","Text":"It\u0027s just e^1x and e^-1x,"},{"Start":"01:36.015 ","End":"01:38.105","Text":"each times the constant."},{"Start":"01:38.105 ","End":"01:41.100","Text":"Putting all the bits together, these two,"},{"Start":"01:41.100 ","End":"01:44.775","Text":"like I said, give me the e^x and e^-x."},{"Start":"01:44.775 ","End":"01:49.560","Text":"These two, the 0 plus or minus 1i, give me this."},{"Start":"01:49.560 ","End":"01:51.780","Text":"I guess I should have put a 1 in here,"},{"Start":"01:51.780 ","End":"01:53.540","Text":"and a 1 in here,"},{"Start":"01:53.540 ","End":"01:55.205","Text":"I put the 0 in here."},{"Start":"01:55.205 ","End":"01:57.710","Text":"Anyway, e_0 is just 1,"},{"Start":"01:57.710 ","End":"02:02.700","Text":"and so we end up with this as our final solution and we\u0027re done."}],"ID":7818},{"Watched":false,"Name":"Exercise 6","Duration":"3m 44s","ChapterTopicVideoID":7744,"CourseChapterTopicPlaylistID":4228,"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":"We have to solve this here differential equation"},{"Start":"00:02.730 ","End":"00:05.685","Text":"just using a bit of a different notation this time,"},{"Start":"00:05.685 ","End":"00:11.025","Text":"instead of y triple prime and y double prime and y prime,"},{"Start":"00:11.025 ","End":"00:13.395","Text":"we\u0027re using the Leibniz notation."},{"Start":"00:13.395 ","End":"00:17.775","Text":"As usual, it\u0027s linear homogeneous constant coefficient."},{"Start":"00:17.775 ","End":"00:21.210","Text":"We know we need to go for the characteristic equation first."},{"Start":"00:21.210 ","End":"00:23.360","Text":"Here it is, but how do we solve it?"},{"Start":"00:23.360 ","End":"00:29.600","Text":"It\u0027s cubic. We\u0027ll use the theorem that if the leading coefficient is 1,"},{"Start":"00:29.600 ","End":"00:36.185","Text":"then any root has got to be a divisor of 20 could be plus or minus."},{"Start":"00:36.185 ","End":"00:40.474","Text":"But in this case there\u0027s a lot of devices actually accounted a dozen."},{"Start":"00:40.474 ","End":"00:47.410","Text":"We just have to go through them one-by-one and substitutes 1 minus 1 2 minus 2 and so on."},{"Start":"00:47.410 ","End":"00:53.375","Text":"First one I found that actually satisfies this as minus 4."},{"Start":"00:53.375 ","End":"00:55.280","Text":"Once we have one root,"},{"Start":"00:55.280 ","End":"01:00.080","Text":"we can then do a long division and reduce the power of this."},{"Start":"01:00.080 ","End":"01:03.200","Text":"I\u0027ll show you the long division."},{"Start":"01:03.200 ","End":"01:06.395","Text":"I\u0027ll give it to you all at once, that of line-by-line."},{"Start":"01:06.395 ","End":"01:09.530","Text":"This is the original polynomial."},{"Start":"01:09.530 ","End":"01:12.390","Text":"Here\u0027s the k minus minus 4,"},{"Start":"01:12.390 ","End":"01:13.960","Text":"which is k plus 4."},{"Start":"01:13.960 ","End":"01:15.965","Text":"I have to divide this into this."},{"Start":"01:15.965 ","End":"01:17.420","Text":"What I do is I say, okay,"},{"Start":"01:17.420 ","End":"01:19.595","Text":"k goes into k^3,"},{"Start":"01:19.595 ","End":"01:25.430","Text":"k^2 times, I take k^2 and I multiply by k plus 4,"},{"Start":"01:25.430 ","End":"01:26.975","Text":"and then I get this."},{"Start":"01:26.975 ","End":"01:28.370","Text":"Then you do the subtraction."},{"Start":"01:28.370 ","End":"01:29.480","Text":"This minus this goes,"},{"Start":"01:29.480 ","End":"01:34.400","Text":"this minus this is minus 2k^2 drops some more terms down."},{"Start":"01:34.400 ","End":"01:37.025","Text":"Then I have a minus 2k^2."},{"Start":"01:37.025 ","End":"01:41.370","Text":"I ask, k into minus 2k^2 goes how many times?"},{"Start":"01:41.370 ","End":"01:45.570","Text":"Minus 2k multiply minus 2k by this, we get this,"},{"Start":"01:45.570 ","End":"01:51.000","Text":"then subtract and k goes into this 5 times and it goes in evenly."},{"Start":"01:51.000 ","End":"01:57.860","Text":"Now this is what we get after the division is k^2 minus 2k plus 5."},{"Start":"01:57.860 ","End":"02:02.915","Text":"What this means is that I can take this original cubic equation"},{"Start":"02:02.915 ","End":"02:08.610","Text":"and write it as k plus 4 times the quadratic equals 0."},{"Start":"02:08.610 ","End":"02:10.715","Text":"Now quadratics we know how to solve."},{"Start":"02:10.715 ","End":"02:18.125","Text":"I mean from here, either k is minus 4 or the quadratic part is 0."},{"Start":"02:18.125 ","End":"02:22.220","Text":"We just have to solve the quadratic usually I just give you the answers here."},{"Start":"02:22.220 ","End":"02:23.765","Text":"I\u0027ll show you with the formula."},{"Start":"02:23.765 ","End":"02:27.505","Text":"Minus b plus or minus the square root of b^2 which is 4,"},{"Start":"02:27.505 ","End":"02:31.940","Text":"minus 4ac is 4 minus 20 is minus 16 over 2a."},{"Start":"02:31.940 ","End":"02:34.010","Text":"Under the square root sign,"},{"Start":"02:34.010 ","End":"02:35.915","Text":"we have a negative number."},{"Start":"02:35.915 ","End":"02:38.720","Text":"Because of the negative we now enter complex numbers,"},{"Start":"02:38.720 ","End":"02:41.240","Text":"so it\u0027s 2 plus or minus the square root of 16 is 4,"},{"Start":"02:41.240 ","End":"02:45.145","Text":"but it have an i here for the minus over 2."},{"Start":"02:45.145 ","End":"02:49.410","Text":"Then we just divide this and get 1 plus or minus 2i."},{"Start":"02:49.410 ","End":"02:52.999","Text":"If I put together all the solutions I found,"},{"Start":"02:52.999 ","End":"02:54.815","Text":"I have minus 4,"},{"Start":"02:54.815 ","End":"02:59.255","Text":"then I have 1 plus or minus 2i."},{"Start":"02:59.255 ","End":"03:02.270","Text":"Now we have to do the second part."},{"Start":"03:02.270 ","End":"03:06.800","Text":"From this, we need to write the solution to the ODE."},{"Start":"03:06.800 ","End":"03:08.180","Text":"For the complex part,"},{"Start":"03:08.180 ","End":"03:15.650","Text":"I\u0027m going to use the formula or whatever that they have a complex conjugate."},{"Start":"03:15.650 ","End":"03:20.330","Text":"A plus or minus bi that contributes this to the solution,"},{"Start":"03:20.330 ","End":"03:22.745","Text":"the minus 4 we know what to do with."},{"Start":"03:22.745 ","End":"03:28.910","Text":"Together, this is what we get from the minus 4 constant times e to the minus 4x."},{"Start":"03:28.910 ","End":"03:31.500","Text":"From this, using this,"},{"Start":"03:31.500 ","End":"03:34.995","Text":"we have e^1x, is just e^x."},{"Start":"03:34.995 ","End":"03:37.425","Text":"Then b is 2 here."},{"Start":"03:37.425 ","End":"03:40.170","Text":"Cos(2x) here, sin(2x) here,"},{"Start":"03:40.170 ","End":"03:42.000","Text":"and another pair of constants."},{"Start":"03:42.000 ","End":"03:44.830","Text":"That\u0027s the answer and we\u0027re done."}],"ID":7819},{"Watched":false,"Name":"Exercise 7","Duration":"4m 3s","ChapterTopicVideoID":7745,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.154","Text":"We have this differential equation to solve and just for a change,"},{"Start":"00:04.154 ","End":"00:09.599","Text":"I thought I\u0027d use the Roman numeral notation for 4th derivative."},{"Start":"00:09.599 ","End":"00:11.055","Text":"You put the Roman IV."},{"Start":"00:11.055 ","End":"00:12.930","Text":"I\u0027ve seen it used sometimes,"},{"Start":"00:12.930 ","End":"00:14.430","Text":"you could write a Roman IV,"},{"Start":"00:14.430 ","End":"00:16.560","Text":"you could put a regular 4 in brackets,"},{"Start":"00:16.560 ","End":"00:18.840","Text":"or you could just put 4 primes,"},{"Start":"00:18.840 ","End":"00:20.895","Text":"all the same just for a change."},{"Start":"00:20.895 ","End":"00:24.915","Text":"Anyway, as usual, we start with the characteristic equation."},{"Start":"00:24.915 ","End":"00:29.595","Text":"I think it\u0027s fairly clear that k^4 plus 1 equals 0,"},{"Start":"00:29.595 ","End":"00:37.980","Text":"and is the formula for finding all the nth roots of a complex number."},{"Start":"00:37.980 ","End":"00:41.705","Text":"I\u0027ll explain it in a moment but what\u0027s important,"},{"Start":"00:41.705 ","End":"00:45.815","Text":"first thing is to write the complex number in polar form,"},{"Start":"00:45.815 ","End":"00:49.490","Text":"which I hope you remember is the form r"},{"Start":"00:49.490 ","End":"00:57.010","Text":"times cosine Theta plus i sine Theta and in our case for minus 1,"},{"Start":"00:57.010 ","End":"01:00.920","Text":"if you imagine the complex plane, the radius,"},{"Start":"01:00.920 ","End":"01:07.760","Text":"like the distance from the origin is 1 and the angle is 180 degrees or Pi."},{"Start":"01:07.760 ","End":"01:10.340","Text":"R is 1, Theta is Pi."},{"Start":"01:10.340 ","End":"01:11.885","Text":"Now back to the formula,"},{"Start":"01:11.885 ","End":"01:17.855","Text":"if I want to know all the nth roots of something in polar form,"},{"Start":"01:17.855 ","End":"01:19.570","Text":"well as n of them,"},{"Start":"01:19.570 ","End":"01:22.695","Text":"I use the letter z, z_1, z_2 up to z_n."},{"Start":"01:22.695 ","End":"01:27.390","Text":"The radius, let me use the right terms r is actually called the"},{"Start":"01:27.390 ","End":"01:32.265","Text":"modulus and I don\u0027t remember if I covered this,"},{"Start":"01:32.265 ","End":"01:36.000","Text":"and the Theta is called the argument."},{"Start":"01:36.000 ","End":"01:39.695","Text":"The polar form is sometimes called the modulus argument form."},{"Start":"01:39.695 ","End":"01:42.065","Text":"Anyway, once we have r and Theta,"},{"Start":"01:42.065 ","End":"01:47.205","Text":"then we have a formula for the all n of the nth roots."},{"Start":"01:47.205 ","End":"01:51.415","Text":"Usually this formula is given with the letter k and not m,"},{"Start":"01:51.415 ","End":"01:53.495","Text":"because we\u0027ve used k already,"},{"Start":"01:53.495 ","End":"02:00.520","Text":"I modified it so that m is the variable and m runs from 0 through n minus 1."},{"Start":"02:00.520 ","End":"02:02.210","Text":"It could run forever,"},{"Start":"02:02.210 ","End":"02:04.085","Text":"but then you\u0027d get repetitions."},{"Start":"02:04.085 ","End":"02:06.395","Text":"These are n different numbers,"},{"Start":"02:06.395 ","End":"02:08.525","Text":"and these will give all the distinct,"},{"Start":"02:08.525 ","End":"02:11.120","Text":"then it just starts repeating itself."},{"Start":"02:11.120 ","End":"02:14.930","Text":"That was a brief review and if I apply it to our case,"},{"Start":"02:14.930 ","End":"02:16.580","Text":"remember r is 1,"},{"Start":"02:16.580 ","End":"02:18.440","Text":"so I\u0027m putting r is 1,"},{"Start":"02:18.440 ","End":"02:19.805","Text":"n is 4,"},{"Start":"02:19.805 ","End":"02:22.325","Text":"Theta is Pi,"},{"Start":"02:22.325 ","End":"02:25.790","Text":"and that\u0027s basically it."},{"Start":"02:25.790 ","End":"02:28.935","Text":"I have to plug in, well,"},{"Start":"02:28.935 ","End":"02:32.145","Text":"really I could say since n is 4,"},{"Start":"02:32.145 ","End":"02:33.888","Text":"that is just 0,"},{"Start":"02:33.888 ","End":"02:35.655","Text":"1, 2, and 3."},{"Start":"02:35.655 ","End":"02:37.850","Text":"Now, if I substitute these,"},{"Start":"02:37.850 ","End":"02:40.580","Text":"I get the following 4 complex numbers,"},{"Start":"02:40.580 ","End":"02:41.960","Text":"the commas separate it."},{"Start":"02:41.960 ","End":"02:44.195","Text":"This would be my k_1,"},{"Start":"02:44.195 ","End":"02:46.790","Text":"this is k_2,"},{"Start":"02:46.790 ","End":"02:51.005","Text":"k_3, and k_4."},{"Start":"02:51.005 ","End":"02:53.495","Text":"Now, I want to organize these."},{"Start":"02:53.495 ","End":"02:58.255","Text":"What I\u0027m going to do is organize them in conjugate pairs,"},{"Start":"02:58.255 ","End":"03:05.595","Text":"the first and the last have the same real part and so do the second, and third."},{"Start":"03:05.595 ","End":"03:08.820","Text":"These 2, I\u0027m going to combine and these 2,"},{"Start":"03:08.820 ","End":"03:10.290","Text":"I\u0027m going to combine,"},{"Start":"03:10.290 ","End":"03:15.615","Text":"so I get root 2 over 2 plus i root 2 and minus i root 2."},{"Start":"03:15.615 ","End":"03:17.310","Text":"The indexing is off."},{"Start":"03:17.310 ","End":"03:20.220","Text":"This will be k_1 and k_2,"},{"Start":"03:20.220 ","End":"03:22.470","Text":"and this will be k_3, k_4."},{"Start":"03:22.470 ","End":"03:24.620","Text":"Now, we have conjugate pairs."},{"Start":"03:24.620 ","End":"03:28.450","Text":"Remember, conjugate pairs a plus or minus bi."},{"Start":"03:28.450 ","End":"03:31.695","Text":"Recall this formula, we\u0027re going to use it twice."},{"Start":"03:31.695 ","End":"03:34.200","Text":"Once with a being root 2 over 2,"},{"Start":"03:34.200 ","End":"03:37.530","Text":"once with a being minus root 2 over 2."},{"Start":"03:37.530 ","End":"03:42.830","Text":"In both cases, b will be root 2 over 2 and so from the first pair,"},{"Start":"03:42.830 ","End":"03:45.350","Text":"this is what I get by substitution,"},{"Start":"03:45.350 ","End":"03:46.970","Text":"for the second pair,"},{"Start":"03:46.970 ","End":"03:48.110","Text":"this is what I get,"},{"Start":"03:48.110 ","End":"03:52.025","Text":"but notice I can\u0027t reuse c_1 and c_2."},{"Start":"03:52.025 ","End":"03:57.100","Text":"I have to use different constants so that it will be c_3, c_4."},{"Start":"03:57.100 ","End":"03:59.265","Text":"All 4 constants are different."},{"Start":"03:59.265 ","End":"04:03.340","Text":"We are done."}],"ID":7820},{"Watched":false,"Name":"Exercise 8","Duration":"2m 13s","ChapterTopicVideoID":7746,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Here we have this differential equation and"},{"Start":"00:03.210 ","End":"00:06.660","Text":"the first thing we do is go to the characteristic."},{"Start":"00:06.660 ","End":"00:11.385","Text":"6 derivative gives us k^6, second derivative, k^2."},{"Start":"00:11.385 ","End":"00:12.990","Text":"This is the equation we have,"},{"Start":"00:12.990 ","End":"00:15.000","Text":"and we\u0027re going to solve it just by factorizing."},{"Start":"00:15.000 ","End":"00:18.495","Text":"First thing is that k^2 goes into both of them."},{"Start":"00:18.495 ","End":"00:20.325","Text":"This is the first step."},{"Start":"00:20.325 ","End":"00:28.575","Text":"Then we can factorize this as a difference of squares and that gives us these 3 factors."},{"Start":"00:28.575 ","End":"00:31.380","Text":"Let\u0027s try setting each 1 to 0."},{"Start":"00:31.380 ","End":"00:33.030","Text":"First, the k^2,"},{"Start":"00:33.030 ","End":"00:34.725","Text":"and that gives us 2 roots,"},{"Start":"00:34.725 ","End":"00:37.650","Text":"k_1 and k_2 are both 0."},{"Start":"00:37.650 ","End":"00:40.495","Text":"Next, just take the k^2 minus 1."},{"Start":"00:40.495 ","End":"00:42.440","Text":"This gives us k^2 equals 1."},{"Start":"00:42.440 ","End":"00:44.360","Text":"K is plus or minus 1."},{"Start":"00:44.360 ","End":"00:47.100","Text":"We\u0027ve got now number 3 and 4,"},{"Start":"00:47.100 ","End":"00:48.480","Text":"1 and minus 1."},{"Start":"00:48.480 ","End":"00:51.090","Text":"Now the last bit if k^2 plus 1 is 0,"},{"Start":"00:51.090 ","End":"00:57.040","Text":"then k^2 is minus 1. k_5 and k_6 are plus i and minus i,"},{"Start":"00:57.040 ","End":"01:03.140","Text":"which I\u0027ll rewrite slightly like this because I wanted in the form of a plus or minus bi."},{"Start":"01:03.140 ","End":"01:06.170","Text":"Continuing, we want to assemble"},{"Start":"01:06.170 ","End":"01:10.025","Text":"all these different ks into a solution to the differential equation."},{"Start":"01:10.025 ","End":"01:14.165","Text":"I\u0027m just putting this here for a reminder when we come to this,"},{"Start":"01:14.165 ","End":"01:15.830","Text":"then we will use this."},{"Start":"01:15.830 ","End":"01:18.035","Text":"But let\u0027s start off with this part here,"},{"Start":"01:18.035 ","End":"01:21.520","Text":"where we have k_1 and k_2 are both 0,"},{"Start":"01:21.520 ","End":"01:25.310","Text":"well, I showed the whole solution,"},{"Start":"01:25.310 ","End":"01:29.390","Text":"but these contribute this part here."},{"Start":"01:29.390 ","End":"01:30.950","Text":"When you have a double root,"},{"Start":"01:30.950 ","End":"01:33.800","Text":"you take e to the power of 1 of them times x,"},{"Start":"01:33.800 ","End":"01:36.470","Text":"and then we have the extra x here."},{"Start":"01:36.470 ","End":"01:38.690","Text":"Notice that because of the double root."},{"Start":"01:38.690 ","End":"01:42.285","Text":"Then the 1 and the minus 1,"},{"Start":"01:42.285 ","End":"01:43.890","Text":"they gave us the next 2 terms,"},{"Start":"01:43.890 ","End":"01:48.210","Text":"e^1x and e to the minus 1x with constants in front."},{"Start":"01:48.210 ","End":"01:51.645","Text":"Finally, the 0 plus or minus 1i,"},{"Start":"01:51.645 ","End":"01:53.895","Text":"here\u0027s where we use this formula."},{"Start":"01:53.895 ","End":"01:55.615","Text":"From these 2,"},{"Start":"01:55.615 ","End":"01:59.030","Text":"we get this except that we can simplify a bit."},{"Start":"01:59.030 ","End":"02:01.370","Text":"Obviously, we don\u0027t need to put the 1 there."},{"Start":"02:01.370 ","End":"02:04.205","Text":"e^0x is just 1."},{"Start":"02:04.205 ","End":"02:07.580","Text":"Oh yeah, we can get rid of this 1 and this minus the 1 here."},{"Start":"02:07.580 ","End":"02:11.375","Text":"Anyway, it all simplifies down to this, obviously."},{"Start":"02:11.375 ","End":"02:13.860","Text":"That\u0027s it. We\u0027re done."}],"ID":7821},{"Watched":false,"Name":"Exercise 9","Duration":"3m 1s","ChapterTopicVideoID":7747,"CourseChapterTopicPlaylistID":4228,"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":"This differential equation is written in operator form,"},{"Start":"00:04.140 ","End":"00:06.435","Text":"the operator D for the differentiation."},{"Start":"00:06.435 ","End":"00:08.670","Text":"You may have seen this before, if you haven\u0027t,"},{"Start":"00:08.670 ","End":"00:11.295","Text":"then let\u0027s just re-interpret it."},{"Start":"00:11.295 ","End":"00:14.985","Text":"I wrote what it means in a way we usually write things."},{"Start":"00:14.985 ","End":"00:17.910","Text":"In any event, we\u0027re going to solve it as usual,"},{"Start":"00:17.910 ","End":"00:22.485","Text":"it\u0027s linear, homogeneous, constant coefficients."},{"Start":"00:22.485 ","End":"00:24.945","Text":"We go for the characteristic equation,"},{"Start":"00:24.945 ","End":"00:26.295","Text":"and this is what we get."},{"Start":"00:26.295 ","End":"00:29.310","Text":"It\u0027s of degree 5. We know it has to have 5 roots,"},{"Start":"00:29.310 ","End":"00:30.930","Text":"but how do we find them?"},{"Start":"00:30.930 ","End":"00:34.570","Text":"Well, remember there\u0027s a theorem that if the leading coefficient here is 1,"},{"Start":"00:34.570 ","End":"00:40.565","Text":"then the solutions have to be whole numbers that are factors of the last term."},{"Start":"00:40.565 ","End":"00:45.015","Text":"For minus 1, it could only be plus or minus 1."},{"Start":"00:45.015 ","End":"00:49.250","Text":"We just try by direct substitution, for example,"},{"Start":"00:49.250 ","End":"00:54.100","Text":"if I put in 1, I got 1 plus 3 plus 2 minus 2 minus 3 minus 1, and it works."},{"Start":"00:54.100 ","End":"00:56.955","Text":"Turns out that minus 1 also works."},{"Start":"00:56.955 ","End":"00:59.130","Text":"That gives us k_1 and k_2,"},{"Start":"00:59.130 ","End":"01:01.440","Text":"but we need 3 more."},{"Start":"01:01.440 ","End":"01:05.605","Text":"The idea is that each of these might repeat."},{"Start":"01:05.605 ","End":"01:07.565","Text":"Since we can\u0027t get any other numbers,"},{"Start":"01:07.565 ","End":"01:09.860","Text":"we check for multiple roots."},{"Start":"01:09.860 ","End":"01:11.420","Text":"The way you do that,"},{"Start":"01:11.420 ","End":"01:14.450","Text":"this theorem that if something\u0027s a double root,"},{"Start":"01:14.450 ","End":"01:18.290","Text":"then it will be a root also of the derivative, and so on."},{"Start":"01:18.290 ","End":"01:19.400","Text":"You can use the second derivative,"},{"Start":"01:19.400 ","End":"01:21.230","Text":"third derivative to see how many times."},{"Start":"01:21.230 ","End":"01:25.490","Text":"Let\u0027s differentiate this and this is what we get."},{"Start":"01:25.490 ","End":"01:29.370","Text":"If you check, 1 doesn\u0027t satisfy this."},{"Start":"01:29.370 ","End":"01:32.625","Text":"I mean 5 plus 12 plus 6 minus 4 minus 3 is not 0,"},{"Start":"01:32.625 ","End":"01:34.320","Text":"but minus 1, if you check,"},{"Start":"01:34.320 ","End":"01:37.305","Text":"does satisfy this equation."},{"Start":"01:37.305 ","End":"01:39.620","Text":"Minus 1 is a double root,"},{"Start":"01:39.620 ","End":"01:41.480","Text":"at least it might be triple or quadruple."},{"Start":"01:41.480 ","End":"01:44.510","Text":"Let\u0027s keep differentiating."},{"Start":"01:44.510 ","End":"01:46.615","Text":"Here\u0027s the next derivative."},{"Start":"01:46.615 ","End":"01:50.450","Text":"Once again, if you plug in minus 1, it will work."},{"Start":"01:50.450 ","End":"01:55.850","Text":"For example, here we\u0027ll get minus 20 and here minus 12,"},{"Start":"01:55.850 ","End":"01:59.850","Text":"that\u0027s minus 32, minus 36 with this,"},{"Start":"01:59.850 ","End":"02:01.635","Text":"and then here we get plus 36."},{"Start":"02:01.635 ","End":"02:04.895","Text":"This is actually at least a triple root now."},{"Start":"02:04.895 ","End":"02:07.130","Text":"Let\u0027s see, once again,"},{"Start":"02:07.130 ","End":"02:09.950","Text":"if that will do it, here\u0027s the next derivative,"},{"Start":"02:09.950 ","End":"02:11.540","Text":"we plug in minus 1,"},{"Start":"02:11.540 ","End":"02:17.155","Text":"we\u0027ve got minus 72 plus 60 plus 12, here 0."},{"Start":"02:17.155 ","End":"02:19.740","Text":"Once again, minus 1."},{"Start":"02:19.740 ","End":"02:27.600","Text":"If we take stock, we had the roots were 1 and then we also had minus 1, 4 times."},{"Start":"02:27.600 ","End":"02:30.005","Text":"When you have this situation,"},{"Start":"02:30.005 ","End":"02:34.730","Text":"the solution to the differential equation is going to be this."},{"Start":"02:34.730 ","End":"02:38.540","Text":"The 1 gives us the e^1x,"},{"Start":"02:38.540 ","End":"02:40.450","Text":"which is just e^x,"},{"Start":"02:40.450 ","End":"02:42.855","Text":"and the minus 1,"},{"Start":"02:42.855 ","End":"02:47.000","Text":"4 times, we have to take first e^x,"},{"Start":"02:47.000 ","End":"02:52.240","Text":"but then we have to throw in an extra x and x^2 and x^3."},{"Start":"02:52.240 ","End":"02:56.030","Text":"When a root repeat, we just keep multiplying by powers of x."},{"Start":"02:56.030 ","End":"03:00.515","Text":"All of this is due to this, and that\u0027s it."},{"Start":"03:00.515 ","End":"03:02.760","Text":"That\u0027s the answer and we\u0027re done."}],"ID":7822},{"Watched":false,"Name":"Exercise 10","Duration":"4m 14s","ChapterTopicVideoID":7748,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"Here we have another linear differential equation, homogeneous,"},{"Start":"00:03.780 ","End":"00:06.000","Text":"constant coefficient so of course,"},{"Start":"00:06.000 ","End":"00:09.300","Text":"the first thing we do is get the characteristic equation."},{"Start":"00:09.300 ","End":"00:10.590","Text":"If you just look at it quickly,"},{"Start":"00:10.590 ","End":"00:13.200","Text":"see that this factorizes,"},{"Start":"00:13.200 ","End":"00:15.330","Text":"it\u0027s one of those special binomial."},{"Start":"00:15.330 ","End":"00:16.695","Text":"Its a perfect square,"},{"Start":"00:16.695 ","End":"00:20.715","Text":"k^4 plus 4 all squared check gives you this."},{"Start":"00:20.715 ","End":"00:22.935","Text":"If this thing squared is 0,"},{"Start":"00:22.935 ","End":"00:28.035","Text":"then k^4 plus 4 must be 0 so k^4 is minus 4."},{"Start":"00:28.035 ","End":"00:30.660","Text":"Later, I\u0027ll return to the fact that this is a square and its"},{"Start":"00:30.660 ","End":"00:33.494","Text":"significant but meanwhile, let\u0027s just continue."},{"Start":"00:33.494 ","End":"00:36.570","Text":"I want to write this minus 4 in polar form,"},{"Start":"00:36.570 ","End":"00:39.208","Text":"which is the modulus is 4,"},{"Start":"00:39.208 ","End":"00:41.390","Text":"the argument is a 180 degrees or Pi,"},{"Start":"00:41.390 ","End":"00:44.735","Text":"just think about where minus 4 is on the complex plane."},{"Start":"00:44.735 ","End":"00:47.465","Text":"It\u0027s a 180 degrees and 4 units from the origin."},{"Start":"00:47.465 ","End":"00:50.300","Text":"The reason I want it in this form is because then I can"},{"Start":"00:50.300 ","End":"00:53.810","Text":"use this result about the nth root."},{"Start":"00:53.810 ","End":"00:58.130","Text":"If I note that something to the power of n is this then there\u0027s"},{"Start":"00:58.130 ","End":"01:03.410","Text":"n different nth roots and they\u0027re given by this formula."},{"Start":"01:03.410 ","End":"01:06.440","Text":"I usually see it with k but I changed"},{"Start":"01:06.440 ","End":"01:09.950","Text":"the k to an m so it wouldn\u0027t clash with the k that we have."},{"Start":"01:09.950 ","End":"01:14.495","Text":"If I substitute r is 4 and Theta is Pi, this is what I get."},{"Start":"01:14.495 ","End":"01:19.560","Text":"Actually, I should have also substituted n as 4 so this is just 0,"},{"Start":"01:19.560 ","End":"01:20.760","Text":"1, 2,"},{"Start":"01:20.760 ","End":"01:22.855","Text":"3 as the 4."},{"Start":"01:22.855 ","End":"01:27.935","Text":"Here also I can write this as a 4 and I need a bit more space here."},{"Start":"01:27.935 ","End":"01:30.860","Text":"Now the 4th root of 4 is like the square root of"},{"Start":"01:30.860 ","End":"01:34.115","Text":"the square root of 4 which is square root of 2."},{"Start":"01:34.115 ","End":"01:36.215","Text":"Let\u0027s look at the first one, for example."},{"Start":"01:36.215 ","End":"01:42.740","Text":"If m is 0 then I get cosin Pi/4 plus isin Pi/4,"},{"Start":"01:42.740 ","End":"01:45.665","Text":"Pi/4 is 45 degrees so this is what I get,"},{"Start":"01:45.665 ","End":"01:49.240","Text":"and then I put m=1, m=2, m=3,"},{"Start":"01:49.240 ","End":"01:53.520","Text":"and I get 4 different complex numbers."},{"Start":"01:53.520 ","End":"01:54.980","Text":"Let\u0027s pair them,"},{"Start":"01:54.980 ","End":"01:56.750","Text":"each one with its conjugate."},{"Start":"01:56.750 ","End":"02:00.040","Text":"This one is the conjugate of this one,"},{"Start":"02:00.040 ","End":"02:02.750","Text":"and this one is the conjugate of this one,"},{"Start":"02:02.750 ","End":"02:06.650","Text":"the same real part and opposite imaginary parts."},{"Start":"02:06.650 ","End":"02:10.655","Text":"I can rewrite this results with a plus or minus sign."},{"Start":"02:10.655 ","End":"02:12.748","Text":"There\u0027s 2 pairs and,"},{"Start":"02:12.748 ","End":"02:14.630","Text":"again, I need more space here."},{"Start":"02:14.630 ","End":"02:17.030","Text":"Now, this simplifies as root 2 times root 2 is 2,"},{"Start":"02:17.030 ","End":"02:19.760","Text":"2/2 is 1 so if you do this,"},{"Start":"02:19.760 ","End":"02:24.815","Text":"we get the results 1 plus or minus i and minus 1 plus or minus i."},{"Start":"02:24.815 ","End":"02:30.510","Text":"Now, back to that matter that we said earlier that it was k^4 plus 4 squared."},{"Start":"02:30.510 ","End":"02:37.160","Text":"That means that each result is actually a double root repeats so"},{"Start":"02:37.160 ","End":"02:44.105","Text":"that really we have 8 roots but the 2nd set of 4 is the same as the 1st set of 4,"},{"Start":"02:44.105 ","End":"02:46.340","Text":"meaning I can go 5,6,7 and 8,"},{"Start":"02:46.340 ","End":"02:48.395","Text":"and these are the same as these means that"},{"Start":"02:48.395 ","End":"02:51.770","Text":"there\u0027s 4 different ones but each one is twice."},{"Start":"02:51.770 ","End":"02:53.710","Text":"Now, let\u0027s see how we interpret this,"},{"Start":"02:53.710 ","End":"02:58.510","Text":"how each one contributes to the general solution of the differential equation."},{"Start":"02:58.510 ","End":"03:03.905","Text":"I want to remind you that when we have a conjugate pair like a plus or minus bi,"},{"Start":"03:03.905 ","End":"03:07.025","Text":"then what this gives is as follows."},{"Start":"03:07.025 ","End":"03:12.480","Text":"In the 1st one, a is 1 and b is 1,"},{"Start":"03:12.480 ","End":"03:15.990","Text":"in this one a is minus 1 and b is 1."},{"Start":"03:15.990 ","End":"03:20.354","Text":"Let\u0027s see, if we take the 1 plus or minus i,"},{"Start":"03:20.354 ","End":"03:23.990","Text":"that will give us this because it\u0027s e^1x,"},{"Start":"03:23.990 ","End":"03:29.600","Text":"that\u0027s the a is 1 and the b is also 1."},{"Start":"03:29.600 ","End":"03:32.234","Text":"But I don\u0027t write cosin 1x, I mean, I could."},{"Start":"03:32.234 ","End":"03:35.570","Text":"Then remember that everything\u0027s duplicate,"},{"Start":"03:35.570 ","End":"03:38.195","Text":"I get another 1 plus or minus i here,"},{"Start":"03:38.195 ","End":"03:45.440","Text":"so I get the same thing except with an extra x thrown in so that would give me this."},{"Start":"03:45.440 ","End":"03:48.590","Text":"But I have to use different constants of cos x c_3 and"},{"Start":"03:48.590 ","End":"03:53.675","Text":"c_4 and now let\u0027s look at the other solutions."},{"Start":"03:53.675 ","End":"03:57.290","Text":"This will give just like this,"},{"Start":"03:57.290 ","End":"04:02.020","Text":"but with a minus 1 because this time i is minus 1 will give this."},{"Start":"04:02.020 ","End":"04:05.810","Text":"Then, because this is repeated we get the same thing"},{"Start":"04:05.810 ","End":"04:10.100","Text":"with an x and multiplied that gives this."},{"Start":"04:10.100 ","End":"04:12.214","Text":"This is all put altogether,"},{"Start":"04:12.214 ","End":"04:15.840","Text":"is our solution and we are done."}],"ID":7823},{"Watched":false,"Name":"Exercise 11","Duration":"2m 35s","ChapterTopicVideoID":7749,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"In this differential equation,"},{"Start":"00:02.010 ","End":"00:04.340","Text":"we\u0027ve used the letter z instead of y."},{"Start":"00:04.340 ","End":"00:07.830","Text":"You shouldn\u0027t get too used to seeing the same letter each time,"},{"Start":"00:07.830 ","End":"00:09.630","Text":"z is a perfectly good letter."},{"Start":"00:09.630 ","End":"00:13.650","Text":"Anyway, it\u0027s linear, homogeneous constant coefficients."},{"Start":"00:13.650 ","End":"00:18.570","Text":"We know that the first step will be to solve the characteristic equation."},{"Start":"00:18.570 ","End":"00:20.340","Text":"That\u0027s fairly clear."},{"Start":"00:20.340 ","End":"00:23.950","Text":"The thing is that this is a cubic equation and there"},{"Start":"00:23.950 ","End":"00:27.750","Text":"is no obvious immediate factoring that I can see,"},{"Start":"00:27.750 ","End":"00:30.570","Text":"so we are going to have to guess solutions."},{"Start":"00:30.570 ","End":"00:34.590","Text":"We\u0027ll use the theorem that when the leading coefficient here is 1,"},{"Start":"00:34.590 ","End":"00:40.970","Text":"then the roots are integer divisors of the free co-efficient, the minus 8."},{"Start":"00:40.970 ","End":"00:42.590","Text":"There is quite a few of them."},{"Start":"00:42.590 ","End":"00:44.191","Text":"All these are possibilities,"},{"Start":"00:44.191 ","End":"00:45.440","Text":"actually 8 of them,"},{"Start":"00:45.440 ","End":"00:49.685","Text":"and we have to plug in each one and see if it\u0027s a root."},{"Start":"00:49.685 ","End":"00:51.290","Text":"For example, plug in 1,"},{"Start":"00:51.290 ","End":"00:55.550","Text":"you got 1 minus 6 plus 12 minus 8 is not 0."},{"Start":"00:55.550 ","End":"01:02.910","Text":"I\u0027ve tried them and actually there is only 1 that works and that is k equals 2."},{"Start":"01:02.910 ","End":"01:05.175","Text":"Let us do a quick mental check."},{"Start":"01:05.175 ","End":"01:07.060","Text":"2^3 is 8,"},{"Start":"01:07.060 ","End":"01:09.725","Text":"well that will cancel with the minus 8,"},{"Start":"01:09.725 ","End":"01:11.640","Text":"k^2 is 4,"},{"Start":"01:11.640 ","End":"01:13.710","Text":"that\u0027s minus 24,"},{"Start":"01:13.710 ","End":"01:16.755","Text":"and plus 12 times 2 is also 24, so those cancel."},{"Start":"01:16.755 ","End":"01:18.240","Text":"We do get 0."},{"Start":"01:18.240 ","End":"01:22.050","Text":"K equals 2 is the only root."},{"Start":"01:22.050 ","End":"01:24.840","Text":"Now, I need 3 roots."},{"Start":"01:24.840 ","End":"01:31.430","Text":"Probably this is a triple root but let\u0027s not take shortcuts."},{"Start":"01:31.430 ","End":"01:36.320","Text":"Let\u0027s just see if it is a double root using the derivative technique."},{"Start":"01:36.320 ","End":"01:39.860","Text":"We take the derivative of this and see if it\u0027s also a root of a derivative."},{"Start":"01:39.860 ","End":"01:42.140","Text":"It\u0027s a double root, and if you keep differentiating,"},{"Start":"01:42.140 ","End":"01:44.530","Text":"it could be triple, quadruple, whatever."},{"Start":"01:44.530 ","End":"01:46.730","Text":"Here\u0027s the first derivative."},{"Start":"01:46.730 ","End":"01:51.590","Text":"Substitute 2, quick calculation shows that this does give 0,"},{"Start":"01:51.590 ","End":"01:54.060","Text":"and so our second root is 2."},{"Start":"01:54.060 ","End":"01:57.230","Text":"I\u0027m betting at the third one is going to be 2 also."},{"Start":"01:57.230 ","End":"01:58.625","Text":"What else could it be?"},{"Start":"01:58.625 ","End":"02:00.365","Text":"Let\u0027s differentiate again."},{"Start":"02:00.365 ","End":"02:02.420","Text":"From here, 6k minus 12,"},{"Start":"02:02.420 ","End":"02:07.815","Text":"and certainly 2 fits here and so 2 is a triple root."},{"Start":"02:07.815 ","End":"02:10.590","Text":"K1, k2, and k3 are all 2."},{"Start":"02:10.590 ","End":"02:12.810","Text":"When this happens, you know what to do."},{"Start":"02:12.810 ","End":"02:16.680","Text":"We have the basic e^2x."},{"Start":"02:16.680 ","End":"02:20.190","Text":"E^2x is here. Well, it\u0027s also here and here."},{"Start":"02:20.190 ","End":"02:21.980","Text":"But when it\u0027s a double root,"},{"Start":"02:21.980 ","End":"02:23.770","Text":"we put an extra x in front."},{"Start":"02:23.770 ","End":"02:26.680","Text":"If it\u0027s a triple root, x^2, and so on."},{"Start":"02:26.680 ","End":"02:29.435","Text":"If it was a quadruple root you put the next cubed."},{"Start":"02:29.435 ","End":"02:35.250","Text":"This is the final solution. We are done."}],"ID":7824},{"Watched":false,"Name":"Exercise 12","Duration":"4m 50s","ChapterTopicVideoID":7750,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"We have to solve this ODE,"},{"Start":"00:02.010 ","End":"00:06.945","Text":"which is homogeneous linear constant coefficients."},{"Start":"00:06.945 ","End":"00:10.365","Text":"The first thing we do is the characteristic equation."},{"Start":"00:10.365 ","End":"00:12.855","Text":"There\u0027s actually more than one way of doing this."},{"Start":"00:12.855 ","End":"00:14.730","Text":"I\u0027m going to do it the long way, but first,"},{"Start":"00:14.730 ","End":"00:17.070","Text":"I\u0027ll do it the short way at the side, then we\u0027ll check."},{"Start":"00:17.070 ","End":"00:19.215","Text":"The short way would just be to say,"},{"Start":"00:19.215 ","End":"00:23.400","Text":"this factorizes into k^2 minus 2,"},{"Start":"00:23.400 ","End":"00:25.890","Text":"k^2 plus 2 = 0."},{"Start":"00:25.890 ","End":"00:27.450","Text":"If k^2 is 2,"},{"Start":"00:27.450 ","End":"00:31.620","Text":"then we get k is plus or minus the square root of 2."},{"Start":"00:31.620 ","End":"00:33.630","Text":"If k^2 is minus 2,"},{"Start":"00:33.630 ","End":"00:38.550","Text":"then we get that k is plus or minus root 2i."},{"Start":"00:38.550 ","End":"00:40.830","Text":"This might be k_1 and 2,"},{"Start":"00:40.830 ","End":"00:44.385","Text":"this might be 3 and 4, and 4 solutions."},{"Start":"00:44.385 ","End":"00:48.575","Text":"I want to practice with taking the nth roots."},{"Start":"00:48.575 ","End":"00:50.745","Text":"Now the long way around,"},{"Start":"00:50.745 ","End":"00:52.755","Text":"but it teaches something."},{"Start":"00:52.755 ","End":"00:54.480","Text":"k^4 equals 4."},{"Start":"00:54.480 ","End":"00:58.890","Text":"I need to find the 4 4th roots of this equation and it\u0027s"},{"Start":"00:58.890 ","End":"01:03.260","Text":"better if we put the 4 in polar form and this is what it is."},{"Start":"01:03.260 ","End":"01:06.410","Text":"This is the modulus and this is the argument."},{"Start":"01:06.410 ","End":"01:12.825","Text":"In general, we write cosine Theta plus i sine Theta."},{"Start":"01:12.825 ","End":"01:14.940","Text":"This cosine Theta is 1,"},{"Start":"01:14.940 ","End":"01:16.380","Text":"sine Theta is 0,"},{"Start":"01:16.380 ","End":"01:18.050","Text":"so this is right."},{"Start":"01:18.050 ","End":"01:19.690","Text":"Now, there\u0027s a formula,"},{"Start":"01:19.690 ","End":"01:21.470","Text":"it\u0027s in complex numbers."},{"Start":"01:21.470 ","End":"01:23.990","Text":"Z is a typical variable for complex."},{"Start":"01:23.990 ","End":"01:26.810","Text":"If I know Z^n,"},{"Start":"01:26.810 ","End":"01:29.060","Text":"I\u0027m thinking polar coordinates,"},{"Start":"01:29.060 ","End":"01:31.865","Text":"then those n roots of this equation,"},{"Start":"01:31.865 ","End":"01:34.640","Text":"Z_1 up to Z_n,"},{"Start":"01:34.640 ","End":"01:41.450","Text":"and what it is is we take the nth root of the modulus r as a positive number,"},{"Start":"01:41.450 ","End":"01:42.920","Text":"just regular nth root,"},{"Start":"01:42.920 ","End":"01:47.435","Text":"and then we have cosine something plus i sine something."},{"Start":"01:47.435 ","End":"01:49.970","Text":"There\u0027s n different possibilities."},{"Start":"01:49.970 ","End":"01:54.935","Text":"We take the Theta from here and the n from here,"},{"Start":"01:54.935 ","End":"01:59.820","Text":"and we let m run from 0 to n minus 1,"},{"Start":"01:59.820 ","End":"02:02.720","Text":"that\u0027s altogether n of these."},{"Start":"02:02.720 ","End":"02:04.880","Text":"You can continue beyond this,"},{"Start":"02:04.880 ","End":"02:06.725","Text":"but then you get repetitions,"},{"Start":"02:06.725 ","End":"02:08.675","Text":"but the n distinct ones are these."},{"Start":"02:08.675 ","End":"02:10.250","Text":"I also want to say that in the books,"},{"Start":"02:10.250 ","End":"02:11.300","Text":"mostly in the formula,"},{"Start":"02:11.300 ","End":"02:12.740","Text":"they use the letter k here,"},{"Start":"02:12.740 ","End":"02:17.985","Text":"but we\u0027ve already used up k so I changed the k to an m; that\u0027s minor point."},{"Start":"02:17.985 ","End":"02:21.060","Text":"Let\u0027s see how this applies in our case."},{"Start":"02:21.060 ","End":"02:23.265","Text":"Our Z is k,"},{"Start":"02:23.265 ","End":"02:28.530","Text":"and here are the 4 solutions gotten by taking m starting from 0,"},{"Start":"02:28.530 ","End":"02:31.710","Text":"4 different values, with 0 to n minus 1 putting in 0,"},{"Start":"02:31.710 ","End":"02:32.865","Text":"1, 2, and 3."},{"Start":"02:32.865 ","End":"02:34.525","Text":"The n here is 4."},{"Start":"02:34.525 ","End":"02:36.440","Text":"The angle Theta is 0,"},{"Start":"02:36.440 ","End":"02:38.485","Text":"so it doesn\u0027t really appear here,"},{"Start":"02:38.485 ","End":"02:42.630","Text":"and here, we have the 4th root of r which is also 4."},{"Start":"02:42.630 ","End":"02:44.280","Text":"Get some more space here."},{"Start":"02:44.280 ","End":"02:48.880","Text":"Now this actually turns out quite simple because 4th root of 4,"},{"Start":"02:48.880 ","End":"02:52.415","Text":"like the square root of the square root of 4 is just the square root of 2."},{"Start":"02:52.415 ","End":"02:54.940","Text":"Now if I put m = 0,"},{"Start":"02:54.940 ","End":"02:58.305","Text":"I get cosine 0 plus i sine 0 is 1."},{"Start":"02:58.305 ","End":"03:00.435","Text":"If you put m =1,"},{"Start":"03:00.435 ","End":"03:05.640","Text":"I\u0027ve got cosine Pi over 2 i sine Pi over 2."},{"Start":"03:05.640 ","End":"03:07.020","Text":"Cosine Pi over 2 is 1,"},{"Start":"03:07.020 ","End":"03:08.400","Text":"sine Pi over 2 is 0,"},{"Start":"03:08.400 ","End":"03:10.310","Text":"so you get i and so on."},{"Start":"03:10.310 ","End":"03:15.410","Text":"Just your basic trigonometry for multiples of 90-degrees or multiples of Pi over 2,"},{"Start":"03:15.410 ","End":"03:19.730","Text":"and we get square root of 2 times these 4 separate numbers in curly braces."},{"Start":"03:19.730 ","End":"03:21.680","Text":"I mean as a set, these 4 numbers,"},{"Start":"03:21.680 ","End":"03:23.900","Text":"root 2 times this, and so on."},{"Start":"03:23.900 ","End":"03:27.080","Text":"Just organize them a bit with plus or minus root 2"},{"Start":"03:27.080 ","End":"03:30.455","Text":"times 1 and root 2 times minus 1 give us this,"},{"Start":"03:30.455 ","End":"03:32.930","Text":"and I\u0027ll call this 1 and 2."},{"Start":"03:32.930 ","End":"03:34.520","Text":"This is my first k,"},{"Start":"03:34.520 ","End":"03:38.270","Text":"this is the second k. Then 3 and 4 from here and here,"},{"Start":"03:38.270 ","End":"03:40.775","Text":"root 2 times i and minus i,"},{"Start":"03:40.775 ","End":"03:43.055","Text":"so it\u0027s plus or minus root 2i."},{"Start":"03:43.055 ","End":"03:46.165","Text":"Now I have the 4 solutions for k,"},{"Start":"03:46.165 ","End":"03:49.277","Text":"and notice that these two are complex conjugates,"},{"Start":"03:49.277 ","End":"03:53.200","Text":"it\u0027s like 0 plus or minus root 2i."},{"Start":"03:53.200 ","End":"03:56.150","Text":"It\u0027s in the form a plus or minus bi,"},{"Start":"03:56.150 ","End":"03:57.800","Text":"because we know what to do with this."},{"Start":"03:57.800 ","End":"04:00.680","Text":"I\u0027m referring to this formula that when we have an a plus or"},{"Start":"04:00.680 ","End":"04:04.370","Text":"minus bi it contributes this to the solution."},{"Start":"04:04.370 ","End":"04:07.970","Text":"That will be these 3 and 4 and that\u0027s for k_1 and 2."},{"Start":"04:07.970 ","End":"04:10.130","Text":"They\u0027re just two different real numbers,"},{"Start":"04:10.130 ","End":"04:13.885","Text":"so each 1 will produce a^k,"},{"Start":"04:13.885 ","End":"04:15.720","Text":"x with a constant in front of it."},{"Start":"04:15.720 ","End":"04:20.120","Text":"In short, the plus root 2 gives me this bit."},{"Start":"04:20.120 ","End":"04:24.095","Text":"The minus root 2 gives me this bit."},{"Start":"04:24.095 ","End":"04:28.300","Text":"Together, the plus or minus root 2i give me,"},{"Start":"04:28.300 ","End":"04:31.480","Text":"well, a is 0, so just 1."},{"Start":"04:31.480 ","End":"04:33.320","Text":"That\u0027s why you don\u0027t see that here."},{"Start":"04:33.320 ","End":"04:35.330","Text":"We get the c, and of course,"},{"Start":"04:35.330 ","End":"04:37.445","Text":"we need different fresh constants,"},{"Start":"04:37.445 ","End":"04:39.523","Text":"so it won\u0027t be 1 and 2, it\u0027ll be 3."},{"Start":"04:39.523 ","End":"04:41.460","Text":"I repeat the 2 here,"},{"Start":"04:41.460 ","End":"04:43.125","Text":"sorry, that\u0027s a 4,"},{"Start":"04:43.125 ","End":"04:44.790","Text":"and b is root 2,"},{"Start":"04:44.790 ","End":"04:48.360","Text":"so it\u0027s cosine root 2x and sine root 2x,"},{"Start":"04:48.360 ","End":"04:50.740","Text":"and that\u0027s the answer."}],"ID":7825},{"Watched":false,"Name":"Exercise 13","Duration":"3m 25s","ChapterTopicVideoID":7751,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"We have here this differential equation to"},{"Start":"00:02.280 ","End":"00:04.260","Text":"solve similar to the ones that we\u0027ve been solving,"},{"Start":"00:04.260 ","End":"00:06.990","Text":"but with a slight difference x is now"},{"Start":"00:06.990 ","End":"00:11.445","Text":"the dependent variable and t is the independent variable."},{"Start":"00:11.445 ","End":"00:17.460","Text":"This is often seen in physics where t is time and x is some distance or displacement."},{"Start":"00:17.460 ","End":"00:19.425","Text":"Anyway, same idea."},{"Start":"00:19.425 ","End":"00:22.320","Text":"We go for the characteristic equation,"},{"Start":"00:22.320 ","End":"00:24.825","Text":"and here it is, and it\u0027s off the sixth degree."},{"Start":"00:24.825 ","End":"00:26.865","Text":"There\u0027s more than one way of solving this."},{"Start":"00:26.865 ","End":"00:30.690","Text":"One idea would be to substitute k^2 equals"},{"Start":"00:30.690 ","End":"00:35.760","Text":"something because you only have even powers of k. That\u0027s one way of doing it."},{"Start":"00:35.760 ","End":"00:39.860","Text":"I\u0027ll just maybe mentioned that you could let k^2 equals t,"},{"Start":"00:39.860 ","End":"00:41.915","Text":"but I\u0027m not going to do it that way."},{"Start":"00:41.915 ","End":"00:46.490","Text":"I\u0027m going to use the theorem that if this leading coefficient is 1,"},{"Start":"00:46.490 ","End":"00:53.225","Text":"then the roots are all divisors of the free coefficient of minus 1."},{"Start":"00:53.225 ","End":"00:55.285","Text":"It has to be 1 or minus 1."},{"Start":"00:55.285 ","End":"00:57.020","Text":"In fact, both of them work."},{"Start":"00:57.020 ","End":"00:58.565","Text":"If you take a quick look, plug-in,"},{"Start":"00:58.565 ","End":"01:03.260","Text":"1 got 1 minus 3 plus 3 minus 1 is certainly 0."},{"Start":"01:03.260 ","End":"01:05.990","Text":"For minus 1 you\u0027ll get the same thing because these are all"},{"Start":"01:05.990 ","End":"01:10.005","Text":"even powers, so same computation."},{"Start":"01:10.005 ","End":"01:12.000","Text":"But we want 6 roots,"},{"Start":"01:12.000 ","End":"01:14.880","Text":"so I\u0027ve only found 2 and they can only be divisor."},{"Start":"01:14.880 ","End":"01:19.550","Text":"The solution to this riddle is that some of these must be multiple roots."},{"Start":"01:19.550 ","End":"01:23.240","Text":"The easiest way to check if something is a multiple root, doubled,"},{"Start":"01:23.240 ","End":"01:27.125","Text":"tripled, whatever is the root of the derivative or second derivative."},{"Start":"01:27.125 ","End":"01:29.075","Text":"Let\u0027s go for the first derivative,"},{"Start":"01:29.075 ","End":"01:30.580","Text":"and here it is."},{"Start":"01:30.580 ","End":"01:33.395","Text":"Let\u0027s see if either of these are roots."},{"Start":"01:33.395 ","End":"01:35.810","Text":"That\u0027s why I set it equal to 0, plug in 1,"},{"Start":"01:35.810 ","End":"01:38.320","Text":"6 minus 12 plus 6,"},{"Start":"01:38.320 ","End":"01:43.280","Text":"0 minus 1 will be the same thing because these are all odd powers."},{"Start":"01:43.280 ","End":"01:44.480","Text":"I plug in minus 1,"},{"Start":"01:44.480 ","End":"01:47.490","Text":"you\u0027ll get minus 0, which is 0."},{"Start":"01:47.490 ","End":"01:49.665","Text":"Both of these are double roots."},{"Start":"01:49.665 ","End":"01:51.840","Text":"That gives me k_3 and k_4."},{"Start":"01:51.840 ","End":"01:53.850","Text":"Now what about k_5 and k_6?"},{"Start":"01:53.850 ","End":"01:56.150","Text":"Let\u0027s try differentiating it again."},{"Start":"01:56.150 ","End":"01:58.350","Text":"Here we are plugging in 1,"},{"Start":"01:58.350 ","End":"02:01.635","Text":"30 minus 36 plus 6 is certainly 0."},{"Start":"02:01.635 ","End":"02:03.620","Text":"Again, only even powers."},{"Start":"02:03.620 ","End":"02:06.500","Text":"This will also give the same thing."},{"Start":"02:06.500 ","End":"02:09.330","Text":"Now we can say that the fifth,"},{"Start":"02:09.330 ","End":"02:11.910","Text":"sixth roots are also 1 and minus 1."},{"Start":"02:11.910 ","End":"02:19.105","Text":"What this means is that this 1 is a triple root because it\u0027s here, here and here."},{"Start":"02:19.105 ","End":"02:23.660","Text":"Minus 1 is also a triple root here, here and here."},{"Start":"02:23.660 ","End":"02:27.125","Text":"Now we know what to do in the case of multiple roots."},{"Start":"02:27.125 ","End":"02:29.150","Text":"I hope you do nothing."},{"Start":"02:29.150 ","End":"02:32.390","Text":"I\u0027ll remind you. Because of the 1,"},{"Start":"02:32.390 ","End":"02:36.755","Text":"basically we get, the starting point is e^t."},{"Start":"02:36.755 ","End":"02:38.090","Text":"Usually it\u0027s t^x."},{"Start":"02:38.090 ","End":"02:42.100","Text":"But remember that we\u0027re now x is a function of t,"},{"Start":"02:42.100 ","End":"02:45.050","Text":"so e^t, it\u0027s a double root."},{"Start":"02:45.050 ","End":"02:48.920","Text":"Then we take this and multiply it by powers usually of x."},{"Start":"02:48.920 ","End":"02:51.230","Text":"But here if t, so we have e^t,"},{"Start":"02:51.230 ","End":"02:54.110","Text":"te^t and t^2e^t,"},{"Start":"02:54.110 ","End":"02:58.265","Text":"where the quadruple root of t^3 e^t and so on."},{"Start":"02:58.265 ","End":"03:00.290","Text":"Each with its constant."},{"Start":"03:00.290 ","End":"03:02.585","Text":"The same thing with the minus 1,"},{"Start":"03:02.585 ","End":"03:07.550","Text":"we have e to the minus 1 t or just e to the minus t. But because"},{"Start":"03:07.550 ","End":"03:12.965","Text":"it\u0027s thrice appears then the next time it has a t tacked onto it,"},{"Start":"03:12.965 ","End":"03:15.830","Text":"and then t^2 tacked onto it."},{"Start":"03:15.830 ","End":"03:21.365","Text":"Of course, the constants will be different constants, 6 different constants."},{"Start":"03:21.365 ","End":"03:23.314","Text":"This is the general solution,"},{"Start":"03:23.314 ","End":"03:26.130","Text":"and we are done."}],"ID":7826},{"Watched":false,"Name":"Exercise 14","Duration":"6m 36s","ChapterTopicVideoID":7752,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"Here we have a third order differential equation,"},{"Start":"00:03.360 ","End":"00:05.669","Text":"linear homogeneous constant coefficient,"},{"Start":"00:05.669 ","End":"00:08.655","Text":"but we also have initial conditions."},{"Start":"00:08.655 ","End":"00:10.830","Text":"Usually when it\u0027s a third order equation,"},{"Start":"00:10.830 ","End":"00:12.300","Text":"you get 3 initial conditions,"},{"Start":"00:12.300 ","End":"00:15.540","Text":"and these enable us to get rid of the constants at the end."},{"Start":"00:15.540 ","End":"00:18.060","Text":"I don\u0027t know why I usually don\u0027t write this,"},{"Start":"00:18.060 ","End":"00:20.775","Text":"but when I see y prime and so on,"},{"Start":"00:20.775 ","End":"00:24.585","Text":"y as a function of x unless otherwise is saying so."},{"Start":"00:24.585 ","End":"00:26.880","Text":"I will also just write IVP,"},{"Start":"00:26.880 ","End":"00:30.795","Text":"I\u0027ve seen in the literature that an initial value problem is sometimes abbreviated."},{"Start":"00:30.795 ","End":"00:34.735","Text":"That means again, differential equation with initial conditions."},{"Start":"00:34.735 ","End":"00:37.160","Text":"But first of all, we\u0027ll solve"},{"Start":"00:37.160 ","End":"00:41.270","Text":"the general equation and at the end we\u0027ll deal with the initial value."},{"Start":"00:41.270 ","End":"00:45.605","Text":"As usual, we want to start with the characteristic equation of this,"},{"Start":"00:45.605 ","End":"00:49.970","Text":"which is this the usual third derivative k^3,"},{"Start":"00:49.970 ","End":"00:52.085","Text":"second derivative k^2, and so on."},{"Start":"00:52.085 ","End":"00:55.090","Text":"It\u0027s a cubic equation, so we want 3 solutions."},{"Start":"00:55.090 ","End":"00:57.478","Text":"There\u0027s more than one way to do this,"},{"Start":"00:57.478 ","End":"01:01.070","Text":"but let\u0027s use a theorem that if we"},{"Start":"01:01.070 ","End":"01:05.240","Text":"have a polynomial with integer coefficients and leading coefficient 1,"},{"Start":"01:05.240 ","End":"01:09.110","Text":"then if it has any whole number solutions,"},{"Start":"01:09.110 ","End":"01:11.540","Text":"they have to be divisors of minus 1."},{"Start":"01:11.540 ","End":"01:17.885","Text":"Let\u0027s try plus or minus 1 and see if either of those will fit."},{"Start":"01:17.885 ","End":"01:19.800","Text":"Plus 1 will actually work,"},{"Start":"01:19.800 ","End":"01:22.650","Text":"you\u0027ll get 1 minus 1 plus 1 minus 1 is 0."},{"Start":"01:22.650 ","End":"01:24.435","Text":"Minus 1 won\u0027t work,"},{"Start":"01:24.435 ","End":"01:27.420","Text":"we\u0027ll get minus 4 here, which is not 0."},{"Start":"01:27.420 ","End":"01:31.640","Text":"At least we have a start, we have k_1=1."},{"Start":"01:31.640 ","End":"01:34.130","Text":"Now we have to find k_2 and k_3."},{"Start":"01:34.130 ","End":"01:39.335","Text":"What we can do is use division of polynomials to help us here,"},{"Start":"01:39.335 ","End":"01:43.295","Text":"because if k=1 is a solution,"},{"Start":"01:43.295 ","End":"01:46.280","Text":"then this is divisible by k minus 1."},{"Start":"01:46.280 ","End":"01:48.245","Text":"Let\u0027s do the long division."},{"Start":"01:48.245 ","End":"01:50.810","Text":"What we do is here we write the,"},{"Start":"01:50.810 ","End":"01:53.740","Text":"this is called the dividend,"},{"Start":"01:53.740 ","End":"01:59.175","Text":"k^3 minus k^2 plus k minus 1."},{"Start":"01:59.175 ","End":"02:02.700","Text":"The divisor is k minus 1,"},{"Start":"02:02.700 ","End":"02:05.044","Text":"and after we divide will get the quotient."},{"Start":"02:05.044 ","End":"02:07.145","Text":"Anyway, you don\u0027t have to worry about those terms."},{"Start":"02:07.145 ","End":"02:09.200","Text":"The way it goes is this,"},{"Start":"02:09.200 ","End":"02:13.520","Text":"k into k^3 goes k^2 times,"},{"Start":"02:13.520 ","End":"02:20.090","Text":"k^2 times this is k^3 minus k^2."},{"Start":"02:20.090 ","End":"02:23.615","Text":"Subtract, this gives us nothing and nothing,"},{"Start":"02:23.615 ","End":"02:27.895","Text":"we get nothing from here, so we just drop down more terms from here."},{"Start":"02:27.895 ","End":"02:29.385","Text":"k minus 1,"},{"Start":"02:29.385 ","End":"02:35.890","Text":"k into k goes 1 times k minus 1 is k minus 1,"},{"Start":"02:35.890 ","End":"02:38.255","Text":"and so that goes in evenly."},{"Start":"02:38.255 ","End":"02:41.135","Text":"I actually prepared a printed out version of this."},{"Start":"02:41.135 ","End":"02:44.270","Text":"I can write this as this times this."},{"Start":"02:44.270 ","End":"02:46.580","Text":"I just wrote it over here next to this."},{"Start":"02:46.580 ","End":"02:48.710","Text":"We\u0027ve already taken care of this,"},{"Start":"02:48.710 ","End":"02:51.230","Text":"k minus 1 is 0, that\u0027s our k_1."},{"Start":"02:51.230 ","End":"02:53.815","Text":"The next possibility is this one,"},{"Start":"02:53.815 ","End":"02:55.890","Text":"and if k^2 plus 1 is 0,"},{"Start":"02:55.890 ","End":"02:57.685","Text":"then k^2 is minus 1."},{"Start":"02:57.685 ","End":"03:00.440","Text":"So we have complex solutions,"},{"Start":"03:00.440 ","End":"03:01.910","Text":"these are actually conjugates."},{"Start":"03:01.910 ","End":"03:06.875","Text":"I could write this as 0 plus or minus 1i."},{"Start":"03:06.875 ","End":"03:12.830","Text":"It\u0027s easier to have it in the form a plus or minus bi in the formula for that."},{"Start":"03:12.830 ","End":"03:15.440","Text":"Let\u0027s collect together what we have,"},{"Start":"03:15.440 ","End":"03:19.130","Text":"k_1 scrolled off the screen was equal to 1."},{"Start":"03:19.130 ","End":"03:23.735","Text":"Now when we have a single solution and not complex,"},{"Start":"03:23.735 ","End":"03:31.215","Text":"then that one means that we get e^1x and there is a constant."},{"Start":"03:31.215 ","End":"03:35.090","Text":"This solution gives me the e^1x,"},{"Start":"03:35.090 ","End":"03:37.475","Text":"and there\u0027s always a constant in front."},{"Start":"03:37.475 ","End":"03:41.420","Text":"When we have an a plus or minus bi solution,"},{"Start":"03:41.420 ","End":"03:49.145","Text":"then that contributes e^ax times 1 constant,"},{"Start":"03:49.145 ","End":"03:55.440","Text":"times cosine bx, plus another constant."},{"Start":"03:55.440 ","End":"03:56.940","Text":"We\u0027ve used c_1,"},{"Start":"03:56.940 ","End":"04:02.685","Text":"so let me say c_2 and c_3, sinbx."},{"Start":"04:02.685 ","End":"04:05.190","Text":"In our case, b is 1,"},{"Start":"04:05.190 ","End":"04:10.320","Text":"so it would be a 1x here and a 1x here."},{"Start":"04:10.320 ","End":"04:18.585","Text":"Then also this disappears because e^0x=1."},{"Start":"04:18.585 ","End":"04:20.270","Text":"This would disappear,"},{"Start":"04:20.270 ","End":"04:22.475","Text":"and in short we get this,"},{"Start":"04:22.475 ","End":"04:24.605","Text":"and of course we don\u0027t need these ones."},{"Start":"04:24.605 ","End":"04:27.785","Text":"That\u0027s the solution to the differential equation."},{"Start":"04:27.785 ","End":"04:29.540","Text":"Remove those ones by the way."},{"Start":"04:29.540 ","End":"04:30.950","Text":"However, we\u0027re not done,"},{"Start":"04:30.950 ","End":"04:34.565","Text":"there is a Part 3 because we had initial conditions."},{"Start":"04:34.565 ","End":"04:37.150","Text":"I should get myself some space here,"},{"Start":"04:37.150 ","End":"04:41.540","Text":"and I copied the initial conditions from back at the top."},{"Start":"04:41.540 ","End":"04:47.029","Text":"What we\u0027ll need is y prime and y double prime in order to substitute 0,"},{"Start":"04:47.029 ","End":"04:50.060","Text":"we already have y. I\u0027ll give you them both at once."},{"Start":"04:50.060 ","End":"04:52.865","Text":"The derivative of e^x is just e^x,"},{"Start":"04:52.865 ","End":"04:54.830","Text":"so this part stays the same."},{"Start":"04:54.830 ","End":"04:59.570","Text":"Cosine, its derivative is minus sine and the derivative of sine is cosine,"},{"Start":"04:59.570 ","End":"05:01.970","Text":"and with sine also derivative is cosine,"},{"Start":"05:01.970 ","End":"05:03.520","Text":"derivative of that minus sine."},{"Start":"05:03.520 ","End":"05:06.094","Text":"This is the first and second derivatives,"},{"Start":"05:06.094 ","End":"05:10.250","Text":"now we can substitute 0 in each of these 3."},{"Start":"05:10.250 ","End":"05:12.380","Text":"In this, here is y,"},{"Start":"05:12.380 ","End":"05:14.540","Text":"here is y prime,"},{"Start":"05:14.540 ","End":"05:16.600","Text":"and here\u0027s y double prime."},{"Start":"05:16.600 ","End":"05:20.615","Text":"The first initial condition y of naught is 3,"},{"Start":"05:20.615 ","End":"05:24.890","Text":"means if I put x=0 here, I get 3."},{"Start":"05:24.890 ","End":"05:31.025","Text":"I\u0027ve got the y of naught is just c_1 times e^0 is 1."},{"Start":"05:31.025 ","End":"05:34.260","Text":"Cosine of 0 is 1,"},{"Start":"05:34.260 ","End":"05:36.915","Text":"and sine is 0,"},{"Start":"05:36.915 ","End":"05:42.975","Text":"so I just get c_1 plus c_2 for y of naught, and that\u0027s 3."},{"Start":"05:42.975 ","End":"05:46.080","Text":"Similarly with y prime and y double prime."},{"Start":"05:46.080 ","End":"05:49.040","Text":"This is 3 equations and 3 unknowns,"},{"Start":"05:49.040 ","End":"05:50.810","Text":"the unknowns are c_1,"},{"Start":"05:50.810 ","End":"05:52.790","Text":"c_2, and c_3."},{"Start":"05:52.790 ","End":"05:55.567","Text":"This part is the system,"},{"Start":"05:55.567 ","End":"05:57.110","Text":"c_1 comes out 1,"},{"Start":"05:57.110 ","End":"05:59.645","Text":"c_2 is 2, and c_3 is 3."},{"Start":"05:59.645 ","End":"06:04.550","Text":"We might take this equation and the last equation and add them."},{"Start":"06:04.550 ","End":"06:07.880","Text":"You\u0027d get 2, c_1=2,"},{"Start":"06:07.880 ","End":"06:09.530","Text":"and so c_1 is 1."},{"Start":"06:09.530 ","End":"06:11.465","Text":"Once we have c_1 is 1,"},{"Start":"06:11.465 ","End":"06:13.995","Text":"from here we get c_2 is 2."},{"Start":"06:13.995 ","End":"06:15.780","Text":"From here, if c_1 is 1,"},{"Start":"06:15.780 ","End":"06:18.240","Text":"then c_3 is 3."},{"Start":"06:18.240 ","End":"06:21.930","Text":"What that does is if we plug in c_1, c_2,"},{"Start":"06:21.930 ","End":"06:25.760","Text":"c_3 here, then this is exactly what we get."},{"Start":"06:25.760 ","End":"06:27.230","Text":"Well, maybe the one here,"},{"Start":"06:27.230 ","End":"06:28.340","Text":"but we don\u0027t need that."},{"Start":"06:28.340 ","End":"06:30.950","Text":"This is the solution to"},{"Start":"06:30.950 ","End":"06:36.720","Text":"the initial value problem and there are no constants to it. We\u0027re done."}],"ID":7827},{"Watched":false,"Name":"Exercise 15","Duration":"7m 13s","ChapterTopicVideoID":7753,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here we have this initial value problem,"},{"Start":"00:03.150 ","End":"00:05.070","Text":"the differential equation part,"},{"Start":"00:05.070 ","End":"00:07.545","Text":"and the initial conditions."},{"Start":"00:07.545 ","End":"00:10.785","Text":"This is a fourth degree, fourth order,"},{"Start":"00:10.785 ","End":"00:12.630","Text":"and a differential equation,"},{"Start":"00:12.630 ","End":"00:15.585","Text":"a linear constant coefficients. I\u0027m a genius."},{"Start":"00:15.585 ","End":"00:18.450","Text":"We go for the characteristic equation,"},{"Start":"00:18.450 ","End":"00:20.145","Text":"and this is it."},{"Start":"00:20.145 ","End":"00:21.780","Text":"Let\u0027s first look for"},{"Start":"00:21.780 ","End":"00:26.340","Text":"some whole number solutions because there\u0027s a theorem that in this case,"},{"Start":"00:26.340 ","End":"00:27.760","Text":"when this coefficient is 1,"},{"Start":"00:27.760 ","End":"00:30.720","Text":"a whole number solution must divide the 8."},{"Start":"00:30.720 ","End":"00:32.700","Text":"These here are all the possibilities."},{"Start":"00:32.700 ","End":"00:35.835","Text":"We just check them 1 by 1, substitute."},{"Start":"00:35.835 ","End":"00:40.970","Text":"Well, I\u0027ve done all that and found that only 2 of them work, 1 and 2."},{"Start":"00:40.970 ","End":"00:44.030","Text":"That\u0027s 2, but we still need 2 more."},{"Start":"00:44.030 ","End":"00:47.660","Text":"We might try to see if any of these are double or triple"},{"Start":"00:47.660 ","End":"00:51.350","Text":"roots by seeing if they\u0027re roots of the derivative."},{"Start":"00:51.350 ","End":"00:54.800","Text":"Save your time, I\u0027ve done that already and they\u0027re not."},{"Start":"00:54.800 ","End":"00:58.445","Text":"These are the only 2 whole number solutions."},{"Start":"00:58.445 ","End":"00:59.995","Text":"What can we do though?"},{"Start":"00:59.995 ","End":"01:02.060","Text":"If 1 is a solution,"},{"Start":"01:02.060 ","End":"01:07.035","Text":"then this thing divides by k-1."},{"Start":"01:07.035 ","End":"01:08.400","Text":"If 2 is a solution,"},{"Start":"01:08.400 ","End":"01:12.005","Text":"then this is divisible by k-2."},{"Start":"01:12.005 ","End":"01:17.225","Text":"Actually, this left-hand side will be divisible by the product of these,"},{"Start":"01:17.225 ","End":"01:20.520","Text":"and I can do a long division. This is the exercise."},{"Start":"01:20.520 ","End":"01:24.155","Text":"I wanted you to take this and divide it by the product of these 2."},{"Start":"01:24.155 ","End":"01:26.375","Text":"Well, first I\u0027ll multiply these 2,"},{"Start":"01:26.375 ","End":"01:28.175","Text":"and that gives me this."},{"Start":"01:28.175 ","End":"01:30.505","Text":"Then I\u0027m just going to tell you the answer,"},{"Start":"01:30.505 ","End":"01:33.220","Text":"comes out as k^2+4."},{"Start":"01:33.220 ","End":"01:35.599","Text":"Well, if you feel cheated,"},{"Start":"01:35.599 ","End":"01:38.480","Text":"I\u0027ll at least show you the calculation,"},{"Start":"01:38.480 ","End":"01:40.625","Text":"the work and you can follow this"},{"Start":"01:40.625 ","End":"01:46.660","Text":"later just longish exercise and we\u0027ve done some long divisions before."},{"Start":"01:46.660 ","End":"01:48.915","Text":"Besides 1 and 2,"},{"Start":"01:48.915 ","End":"01:51.105","Text":"we now are looking for 2 other solutions."},{"Start":"01:51.105 ","End":"01:54.735","Text":"Those will be the solutions of k^2+4=0,"},{"Start":"01:54.735 ","End":"01:56.100","Text":"and this we know how to do."},{"Start":"01:56.100 ","End":"01:57.960","Text":"They\u0027ll be complex solutions."},{"Start":"01:57.960 ","End":"01:59.640","Text":"We bring the 4 to the other side,"},{"Start":"01:59.640 ","End":"02:01.575","Text":"say k^2 is -4,"},{"Start":"02:01.575 ","End":"02:06.510","Text":"and then k_3 and k_4 are +2i and -2i."},{"Start":"02:06.510 ","End":"02:08.310","Text":"Since they\u0027re off the screen,"},{"Start":"02:08.310 ","End":"02:13.215","Text":"I want to remind you that k_1 was 1 and k_2 was 2."},{"Start":"02:13.215 ","End":"02:15.425","Text":"Putting all this stuff together,"},{"Start":"02:15.425 ","End":"02:19.290","Text":"what we get is from the Solution 1,"},{"Start":"02:19.290 ","End":"02:21.840","Text":"we get e^x,"},{"Start":"02:21.840 ","End":"02:23.430","Text":"I didn\u0027t write the 1,"},{"Start":"02:23.430 ","End":"02:25.185","Text":"and from 2,"},{"Start":"02:25.185 ","End":"02:28.770","Text":"we get e^2x also with its constant."},{"Start":"02:28.770 ","End":"02:33.214","Text":"When you have a complex conjugate pair like this,"},{"Start":"02:33.214 ","End":"02:37.580","Text":"I won\u0027t write the general formula again for a plus or minus bi,"},{"Start":"02:37.580 ","End":"02:40.400","Text":"but because the real part is 0,"},{"Start":"02:40.400 ","End":"02:46.650","Text":"normally there would be an e^ax times all of this,"},{"Start":"02:46.650 ","End":"02:48.735","Text":"but if a is 0,"},{"Start":"02:48.735 ","End":"02:50.130","Text":"then this comes out 1,"},{"Start":"02:50.130 ","End":"02:52.320","Text":"so we don\u0027t get the e^ax part."},{"Start":"02:52.320 ","End":"02:56.100","Text":"We just get the cosine bx and the sine bx,"},{"Start":"02:56.100 ","End":"02:57.690","Text":"and the b is 2."},{"Start":"02:57.690 ","End":"03:00.500","Text":"Well, we\u0027ve seen this plenty of times before,"},{"Start":"03:00.500 ","End":"03:06.970","Text":"so I\u0027ll just say that this plus or minus 2i is what gives us this as a pair."},{"Start":"03:06.970 ","End":"03:10.010","Text":"That\u0027s the general solution, but remember,"},{"Start":"03:10.010 ","End":"03:13.220","Text":"this is an initial value problem when we have initial values."},{"Start":"03:13.220 ","End":"03:17.190","Text":"Let me just copy this on the next page, and that\u0027s here."},{"Start":"03:17.190 ","End":"03:21.575","Text":"Also in Step 3 now I\u0027m going to use the initial conditions."},{"Start":"03:21.575 ","End":"03:23.060","Text":"If you go back to the beginning,"},{"Start":"03:23.060 ","End":"03:25.790","Text":"you see that I just copied them as they were."},{"Start":"03:25.790 ","End":"03:28.700","Text":"We\u0027ve got 4 conditions for y,"},{"Start":"03:28.700 ","End":"03:31.670","Text":"y\u0027, y\", and y\u0027\u0027\u0027."},{"Start":"03:31.670 ","End":"03:35.985","Text":"Their values when x is 0 are as given."},{"Start":"03:35.985 ","End":"03:43.115","Text":"We\u0027ll actually need to differentiate this 3 times to get y\u0027, y\", y\"\u0027."},{"Start":"03:43.115 ","End":"03:45.590","Text":"They\u0027ll give you them all 3 at once."},{"Start":"03:45.590 ","End":"03:48.780","Text":"It\u0027s so standard. I\u0027m not going into go into all the details."},{"Start":"03:48.780 ","End":"03:51.825","Text":"We\u0027re speeding things up a bit in this exercise."},{"Start":"03:51.825 ","End":"03:54.930","Text":"Wait a minute."},{"Start":"03:54.930 ","End":"04:01.370","Text":"I need to keep y because now I want to substitute 0 in each of the derivatives,"},{"Start":"04:01.370 ","End":"04:03.320","Text":"just putting a mark to say,"},{"Start":"04:03.320 ","End":"04:06.215","Text":"I\u0027m going to substitute x=0 in all of these."},{"Start":"04:06.215 ","End":"04:07.830","Text":"Now when I put x is 0,"},{"Start":"04:07.830 ","End":"04:09.915","Text":"2x is also going to be 0."},{"Start":"04:09.915 ","End":"04:13.440","Text":"The results I need are that e^0 is 1."},{"Start":"04:13.440 ","End":"04:17.655","Text":"I need to know that cosine 0 is also 1,"},{"Start":"04:17.655 ","End":"04:20.775","Text":"but sine 0 is 0."},{"Start":"04:20.775 ","End":"04:24.455","Text":"Now I can make these substitutions."},{"Start":"04:24.455 ","End":"04:27.710","Text":"I can make x=0 in the 1 over here,"},{"Start":"04:27.710 ","End":"04:28.955","Text":"and in these 3,"},{"Start":"04:28.955 ","End":"04:35.520","Text":"and we get this system of 4 equations in 4 unknowns,"},{"Start":"04:35.520 ","End":"04:38.790","Text":"c_1, c_2, c_3, c_4."},{"Start":"04:38.790 ","End":"04:40.875","Text":"This is the part that interests me now."},{"Start":"04:40.875 ","End":"04:42.375","Text":"Now it\u0027s a straightforward,"},{"Start":"04:42.375 ","End":"04:44.645","Text":"well, maybe not so straightforward."},{"Start":"04:44.645 ","End":"04:48.780","Text":"Well, straightforward, they\u0027re not so easy in linear algebra."},{"Start":"04:48.780 ","End":"04:51.770","Text":"You\u0027re going to have to remember some results from linear algebra."},{"Start":"04:51.770 ","End":"04:54.350","Text":"I hope you do or that you\u0027ve learned it."},{"Start":"04:54.350 ","End":"04:56.495","Text":"I\u0027m going to proceed anyway."},{"Start":"04:56.495 ","End":"04:58.580","Text":"We\u0027re going to use matrices."},{"Start":"04:58.580 ","End":"05:02.944","Text":"This is the augmented matrix I get from this system."},{"Start":"05:02.944 ","End":"05:05.660","Text":"Here I have a missing c_4 that\u0027s 0,"},{"Start":"05:05.660 ","End":"05:07.715","Text":"so 1 1 1 0."},{"Start":"05:07.715 ","End":"05:10.995","Text":"Here 1 2, missing c_3."},{"Start":"05:10.995 ","End":"05:16.280","Text":"There\u0027s a 0 and then 2 and so on after this vertical line, the right-hand sides."},{"Start":"05:16.280 ","End":"05:22.640","Text":"Our task now is to bring this matrix into echelon form using row operations."},{"Start":"05:22.640 ","End":"05:30.180","Text":"What I can do first is subtract the first row from the other 3 to get 0s here,"},{"Start":"05:30.180 ","End":"05:33.140","Text":"and that gives us this matrix."},{"Start":"05:33.140 ","End":"05:34.490","Text":"I\u0027ll just give you an example."},{"Start":"05:34.490 ","End":"05:37.580","Text":"The second row, the 1-1 is 0,"},{"Start":"05:37.580 ","End":"05:40.235","Text":"2-1 is 1, 0-1,"},{"Start":"05:40.235 ","End":"05:43.655","Text":"-1, 2-0, and 5-2."},{"Start":"05:43.655 ","End":"05:46.150","Text":"From each row, we subtract the first."},{"Start":"05:46.150 ","End":"05:47.820","Text":"This is what we have now."},{"Start":"05:47.820 ","End":"05:50.705","Text":"Continuing, here\u0027s the next form."},{"Start":"05:50.705 ","End":"05:55.265","Text":"I got this by trying to get 0s below this 1."},{"Start":"05:55.265 ","End":"05:58.775","Text":"I subtract 3 times this row from this row,"},{"Start":"05:58.775 ","End":"06:02.650","Text":"and I subtract 7 times this row from this row,"},{"Start":"06:02.650 ","End":"06:05.120","Text":"and if you do the computations,"},{"Start":"06:05.120 ","End":"06:06.664","Text":"we should get this."},{"Start":"06:06.664 ","End":"06:14.480","Text":"Next. I made 0 here by adding 3 times this row to this row."},{"Start":"06:14.480 ","End":"06:17.495","Text":"3 times -2+6 is 0."},{"Start":"06:17.495 ","End":"06:22.790","Text":"3 times -6 is -18, +-22 is 40."},{"Start":"06:22.790 ","End":"06:27.350","Text":"3 times -30 is 90 plus this,"},{"Start":"06:27.350 ","End":"06:28.835","Text":"-160."},{"Start":"06:28.835 ","End":"06:30.545","Text":"Let\u0027s keep going."},{"Start":"06:30.545 ","End":"06:34.049","Text":"Since this is already in echelon form, we\u0027ve got steps,"},{"Start":"06:34.049 ","End":"06:37.580","Text":"it\u0027s time to go back to the variables c_1,"},{"Start":"06:37.580 ","End":"06:39.110","Text":"c_2, c_3, c_4,"},{"Start":"06:39.110 ","End":"06:41.120","Text":"and that gives us this equation."},{"Start":"06:41.120 ","End":"06:44.195","Text":"Then we solve it from bottom up."},{"Start":"06:44.195 ","End":"06:50.930","Text":"For example, from here we divide by -40 and we get c_4=4."},{"Start":"06:50.930 ","End":"06:53.675","Text":"Then we plug it into here and get c_3,"},{"Start":"06:53.675 ","End":"06:55.700","Text":"and then we have c_4 and c_3 here,"},{"Start":"06:55.700 ","End":"06:57.730","Text":"so we get c_2 and so on."},{"Start":"06:57.730 ","End":"07:00.210","Text":"Here they are in the order we got them."},{"Start":"07:00.210 ","End":"07:01.640","Text":"Now that we have the c\u0027s,"},{"Start":"07:01.640 ","End":"07:04.860","Text":"we just have to substitute in the general solution."},{"Start":"07:04.860 ","End":"07:05.870","Text":"Where there was c_1,"},{"Start":"07:05.870 ","End":"07:06.890","Text":"c_2, c_3,"},{"Start":"07:06.890 ","End":"07:09.485","Text":"c_4, we just put the values here."},{"Start":"07:09.485 ","End":"07:11.120","Text":"This is our answer,"},{"Start":"07:11.120 ","End":"07:13.590","Text":"and we\u0027re finally done."}],"ID":7828},{"Watched":false,"Name":"Exercise 16","Duration":"4m 35s","ChapterTopicVideoID":7754,"CourseChapterTopicPlaylistID":4228,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Now here\u0027s a strange looking exercise."},{"Start":"00:02.580 ","End":"00:07.515","Text":"We\u0027re given an ordinary differential equation of the 6th order, homogeneous."},{"Start":"00:07.515 ","End":"00:11.625","Text":"I should have said also linear, with constant coefficients."},{"Start":"00:11.625 ","End":"00:18.510","Text":"Given that one of the solutions of this equation is x^2 e^x cosine 2x,"},{"Start":"00:18.510 ","End":"00:20.490","Text":"we have to find a,"},{"Start":"00:20.490 ","End":"00:23.168","Text":"the general solution of the equation,"},{"Start":"00:23.168 ","End":"00:26.115","Text":"and then the equation itself."},{"Start":"00:26.115 ","End":"00:31.380","Text":"Let\u0027s start with part a. When we see cosines and sines,"},{"Start":"00:31.380 ","End":"00:35.985","Text":"it means that it came from a complex numbers with conjugate pairs."},{"Start":"00:35.985 ","End":"00:40.605","Text":"a plus bi normally gives us something of this form."},{"Start":"00:40.605 ","End":"00:44.749","Text":"However, it could be that this is a multiple,"},{"Start":"00:44.749 ","End":"00:46.775","Text":"like a double or triple root."},{"Start":"00:46.775 ","End":"00:48.215","Text":"If that was the case,"},{"Start":"00:48.215 ","End":"00:50.570","Text":"then we\u0027d have to modify this by,"},{"Start":"00:50.570 ","End":"00:52.805","Text":"if it was double root, we\u0027d put x,"},{"Start":"00:52.805 ","End":"00:55.535","Text":"if it was a triple root, we\u0027d put x^2."},{"Start":"00:55.535 ","End":"01:00.740","Text":"In fact, it has to be a triple root because we see that we do have an x^2 here."},{"Start":"01:00.740 ","End":"01:03.410","Text":"But also, when we have a cosine,"},{"Start":"01:03.410 ","End":"01:05.240","Text":"we also have a sine with it."},{"Start":"01:05.240 ","End":"01:09.665","Text":"Now when we have this x^2 in this case or x^n in general,"},{"Start":"01:09.665 ","End":"01:13.220","Text":"it also comes with the 1 and the x."},{"Start":"01:13.220 ","End":"01:14.345","Text":"In other words, we have,"},{"Start":"01:14.345 ","End":"01:16.370","Text":"if this thing was,"},{"Start":"01:16.370 ","End":"01:18.570","Text":"just call it box,"},{"Start":"01:18.570 ","End":"01:22.125","Text":"then we\u0027d have 1 times box,"},{"Start":"01:22.125 ","End":"01:25.080","Text":"plus x times box,"},{"Start":"01:25.080 ","End":"01:27.480","Text":"plus x^2 times box."},{"Start":"01:27.480 ","End":"01:29.700","Text":"We don\u0027t skip powers of x."},{"Start":"01:29.700 ","End":"01:33.750","Text":"That means that if x^2 e^x cos2x,"},{"Start":"01:33.750 ","End":"01:37.058","Text":"then also 1 and x times this,"},{"Start":"01:37.058 ","End":"01:41.130","Text":"are also solutions and that\u0027s summarized here."},{"Start":"01:41.130 ","End":"01:45.000","Text":"x^2 times this, x times this,"},{"Start":"01:45.000 ","End":"01:47.465","Text":"and just 1 times this."},{"Start":"01:47.465 ","End":"01:50.300","Text":"Now remember also that I said that when there\u0027s a cosine,"},{"Start":"01:50.300 ","End":"01:53.625","Text":"there\u0027s also a sine because they come in pairs,"},{"Start":"01:53.625 ","End":"01:55.655","Text":"and so I\u0027ve summarized that here,"},{"Start":"01:55.655 ","End":"01:58.685","Text":"they come in pairs of conjugates."},{"Start":"01:58.685 ","End":"02:01.490","Text":"We also have the same as these 3,"},{"Start":"02:01.490 ","End":"02:04.675","Text":"but with a sine instead of a cosine."},{"Start":"02:04.675 ","End":"02:07.610","Text":"Out of this one solution, we\u0027ve suddenly got 6."},{"Start":"02:07.610 ","End":"02:08.930","Text":"We\u0027ve got 1, 2, 3,"},{"Start":"02:08.930 ","End":"02:11.300","Text":"and then we have these 4,"},{"Start":"02:11.300 ","End":"02:13.115","Text":"5, and 6."},{"Start":"02:13.115 ","End":"02:18.155","Text":"These are just the basics that we have to combine with constants."},{"Start":"02:18.155 ","End":"02:19.705","Text":"The general solution,"},{"Start":"02:19.705 ","End":"02:25.205","Text":"and if I organize it also taking the e^x out of the brackets from this 1 and this 1,"},{"Start":"02:25.205 ","End":"02:28.084","Text":"and throwing in 6 different constants,"},{"Start":"02:28.084 ","End":"02:30.440","Text":"this is how it will look."},{"Start":"02:30.440 ","End":"02:31.925","Text":"This and this,"},{"Start":"02:31.925 ","End":"02:33.410","Text":"no the x^2 with the x^2."},{"Start":"02:33.410 ","End":"02:36.620","Text":"Anyway, you get the idea and we have c_1 through c_6."},{"Start":"02:36.620 ","End":"02:38.840","Text":"Now let\u0027s move on to part B,"},{"Start":"02:38.840 ","End":"02:40.835","Text":"where we have to find the equation,"},{"Start":"02:40.835 ","End":"02:43.100","Text":"and let\u0027s see how we do that."},{"Start":"02:43.100 ","End":"02:44.660","Text":"We know what k is,"},{"Start":"02:44.660 ","End":"02:49.400","Text":"because we know what a and b are in this a plus or minus bi."},{"Start":"02:49.400 ","End":"02:53.960","Text":"The area is the coefficient e to the power of something x."},{"Start":"02:53.960 ","End":"02:56.840","Text":"The 1 is not explicitly written,"},{"Start":"02:56.840 ","End":"02:59.825","Text":"but this is e^1x everywhere."},{"Start":"02:59.825 ","End":"03:04.765","Text":"Everywhere here I could have written e^1x,"},{"Start":"03:04.765 ","End":"03:09.390","Text":"so a is 1 and the 2 gives me that b is 2."},{"Start":"03:09.390 ","End":"03:12.420","Text":"Our k, k_1 and k_2,"},{"Start":"03:12.420 ","End":"03:15.220","Text":"1 plus or minus 2i."},{"Start":"03:16.400 ","End":"03:18.530","Text":"I\u0027m going to run out of space."},{"Start":"03:18.530 ","End":"03:19.940","Text":"I need some more space."},{"Start":"03:19.940 ","End":"03:24.395","Text":"I\u0027m heading for an equation that this solves."},{"Start":"03:24.395 ","End":"03:26.140","Text":"Here\u0027s a few steps."},{"Start":"03:26.140 ","End":"03:28.665","Text":"Bring the 1 to the left-hand side."},{"Start":"03:28.665 ","End":"03:30.300","Text":"Now square this,"},{"Start":"03:30.300 ","End":"03:32.085","Text":"k minus 1^2 is this,"},{"Start":"03:32.085 ","End":"03:34.695","Text":"and 2i^2 is minus 4."},{"Start":"03:34.695 ","End":"03:36.900","Text":"Then of course I bring the minus 4 to the left,"},{"Start":"03:36.900 ","End":"03:42.065","Text":"so this is the equation that 1 plus or minus 2i solves."},{"Start":"03:42.065 ","End":"03:47.240","Text":"But remember, we know that it was a triple solution because we had the 1x,"},{"Start":"03:47.240 ","End":"03:50.310","Text":"x^2 thing. This is what I write."},{"Start":"03:50.310 ","End":"03:56.825","Text":"I mean, normally I would have written k^2 minus 2_k plus 5 cubed."},{"Start":"03:56.825 ","End":"03:58.910","Text":"Certainly it would have been more compact,"},{"Start":"03:58.910 ","End":"04:01.220","Text":"but I intend to multiply it out,"},{"Start":"04:01.220 ","End":"04:02.720","Text":"so I wrote it this way,"},{"Start":"04:02.720 ","End":"04:06.095","Text":"instead of the power of 3 it\u0027s 3 factors."},{"Start":"04:06.095 ","End":"04:09.680","Text":"I\u0027m definitely not going to spend time doing this multiplication."},{"Start":"04:09.680 ","End":"04:15.640","Text":"This is what the result that I get and I\u0027ll leave it for you to check on your own."},{"Start":"04:15.640 ","End":"04:18.290","Text":"This is going to be the characteristic equation."},{"Start":"04:18.290 ","End":"04:22.145","Text":"Now to get from the characteristic equation back to the ODE,"},{"Start":"04:22.145 ","End":"04:27.140","Text":"we just replace k^6th by y 6th derivative and so on."},{"Start":"04:27.140 ","End":"04:30.215","Text":"K to the 5th is the 5th derivative."},{"Start":"04:30.215 ","End":"04:32.165","Text":"K^2, second derivative."},{"Start":"04:32.165 ","End":"04:36.150","Text":"This is the equation, and we\u0027re done."}],"ID":7829}],"Thumbnail":null,"ID":4228},{"Name":"Method of Undetermined Coefficients","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"4m 13s","ChapterTopicVideoID":7735,"CourseChapterTopicPlaylistID":4229,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"Here we have a non-homogeneous linear differential equation"},{"Start":"00:04.740 ","End":"00:07.050","Text":"constant coefficients of the third-order."},{"Start":"00:07.050 ","End":"00:13.140","Text":"A non-homogeneous; we\u0027re going to choose the method of undetermined coefficients,"},{"Start":"00:13.140 ","End":"00:14.595","Text":"there\u0027s more than one method,"},{"Start":"00:14.595 ","End":"00:20.850","Text":"and any event we always start with solving the homogeneous equation."},{"Start":"00:20.850 ","End":"00:25.400","Text":"Homogeneous of course means when there\u0027s a 0 on the right instead of whatever\u0027s there."},{"Start":"00:25.400 ","End":"00:28.970","Text":"We go to the characteristic equation, this is routine."},{"Start":"00:28.970 ","End":"00:33.530","Text":"A triple prime is replaced by k^3 y\u0027\u0027 k^2,"},{"Start":"00:33.530 ","End":"00:35.590","Text":"single prime is k,"},{"Start":"00:35.590 ","End":"00:38.660","Text":"and there was just the y would go with 1."},{"Start":"00:38.660 ","End":"00:41.450","Text":"Now this is a cubic equation,"},{"Start":"00:41.450 ","End":"00:44.180","Text":"but we can solve it because there\u0027s a missing constant term,"},{"Start":"00:44.180 ","End":"00:45.755","Text":"so we can factorize,"},{"Start":"00:45.755 ","End":"00:48.800","Text":"and now we\u0027ve got it down to either k is 0,"},{"Start":"00:48.800 ","End":"00:50.884","Text":"or this quadratic is 0."},{"Start":"00:50.884 ","End":"00:52.250","Text":"The solution of the quadratic,"},{"Start":"00:52.250 ","End":"00:55.250","Text":"I\u0027m just going give you there -1 and 3,"},{"Start":"00:55.250 ","End":"00:57.620","Text":"and of course there\u0027s the 0 from here."},{"Start":"00:57.620 ","End":"01:04.130","Text":"These are the 3 roots of this characteristic equation, 0,-1, and 3,"},{"Start":"01:04.130 ","End":"01:08.450","Text":"and that will give us the general solution to the homogeneous,"},{"Start":"01:08.450 ","End":"01:11.825","Text":"where we just take a linear combination of e^kx,"},{"Start":"01:11.825 ","End":"01:17.000","Text":"where k is 0 or -1, or 3."},{"Start":"01:17.000 ","End":"01:18.680","Text":"Let me put the constant in front,"},{"Start":"01:18.680 ","End":"01:21.205","Text":"that\u0027s what linear combination means."},{"Start":"01:21.205 ","End":"01:24.450","Text":"Of course I don\u0027t write it to the 0 x because that\u0027s just 1,"},{"Start":"01:24.450 ","End":"01:27.605","Text":"so this is the general solution of the homogeneous,"},{"Start":"01:27.605 ","End":"01:29.545","Text":"that was the first step."},{"Start":"01:29.545 ","End":"01:32.505","Text":"In step 2, we look for a private solution,"},{"Start":"01:32.505 ","End":"01:37.790","Text":"let me remind you what the right-hand side of the differential equation."},{"Start":"01:37.790 ","End":"01:43.190","Text":"Was 2 sine x minus 4 cosine x."},{"Start":"01:43.190 ","End":"01:46.550","Text":"What we do is we take each piece and keep"},{"Start":"01:46.550 ","End":"01:52.345","Text":"differentiating it until we get nothing new and ignoring constants."},{"Start":"01:52.345 ","End":"01:56.040","Text":"The derivative of 2 sine x is 2 cosine x,"},{"Start":"01:56.040 ","End":"01:58.650","Text":"the derivative of that is -2 sine x,"},{"Start":"01:58.650 ","End":"01:59.810","Text":"and we have that already,"},{"Start":"01:59.810 ","End":"02:01.550","Text":"as I say, constant is not important."},{"Start":"02:01.550 ","End":"02:04.850","Text":"Similarly, we don\u0027t even have to bother with 4 cosine x,"},{"Start":"02:04.850 ","End":"02:08.240","Text":"or, -4 because the cosine x is already in the list."},{"Start":"02:08.240 ","End":"02:11.075","Text":"Once we have the list of basic functions,"},{"Start":"02:11.075 ","End":"02:14.570","Text":"then our particular solution that we\u0027re looking"},{"Start":"02:14.570 ","End":"02:18.440","Text":"for will be of the form 1 constant times this,"},{"Start":"02:18.440 ","End":"02:19.940","Text":"plus another constant times this,"},{"Start":"02:19.940 ","End":"02:24.155","Text":"however many they are we at constant times whatever it is."},{"Start":"02:24.155 ","End":"02:27.845","Text":"Now this Y_p has to satisfy the differential equation,"},{"Start":"02:27.845 ","End":"02:30.880","Text":"so we\u0027ll need derivatives up to the third order,"},{"Start":"02:30.880 ","End":"02:33.650","Text":"and these are straightforward enough to compute,"},{"Start":"02:33.650 ","End":"02:35.345","Text":"I just presented them."},{"Start":"02:35.345 ","End":"02:39.650","Text":"Now we need to substitute these in the ODE,"},{"Start":"02:39.650 ","End":"02:41.765","Text":"here it is to remind you."},{"Start":"02:41.765 ","End":"02:44.305","Text":"Where we see y\u0027\u0027\u0027,"},{"Start":"02:44.305 ","End":"02:48.270","Text":"we put this, just call it y, instead of y_p."},{"Start":"02:48.270 ","End":"02:52.315","Text":"Instead of y\u0027\u0027 we put this y\u0027,"},{"Start":"02:52.315 ","End":"02:56.645","Text":"we put this, and this is the left-hand side."},{"Start":"02:56.645 ","End":"03:01.190","Text":"On the right you just copy this 2 sine x minus 4 sine x."},{"Start":"03:01.190 ","End":"03:03.490","Text":"Now we want to simplify thing."},{"Start":"03:03.490 ","End":"03:07.655","Text":"If you expand the left-hand side and collect together sines and cosine,"},{"Start":"03:07.655 ","End":"03:09.860","Text":"this is what you get for sine,"},{"Start":"03:09.860 ","End":"03:11.570","Text":"this is what you get for cosine."},{"Start":"03:11.570 ","End":"03:15.390","Text":"Now, these 2 functions have to be the same,"},{"Start":"03:15.390 ","End":"03:17.805","Text":"it\u0027s not an equation in x or anything,"},{"Start":"03:17.805 ","End":"03:19.995","Text":"these have to be the same function."},{"Start":"03:19.995 ","End":"03:24.320","Text":"What we\u0027re saying is that this number is got to equal 2,"},{"Start":"03:24.320 ","End":"03:27.725","Text":"and this number is got to equal -4,"},{"Start":"03:27.725 ","End":"03:30.550","Text":"so we get 2 equations."},{"Start":"03:30.550 ","End":"03:32.490","Text":"Here we are, this is equal to 2,"},{"Start":"03:32.490 ","End":"03:34.590","Text":"this is equal to -4."},{"Start":"03:34.590 ","End":"03:36.060","Text":"I\u0027ll just give you the solutions,"},{"Start":"03:36.060 ","End":"03:37.340","Text":"you know how to solve this."},{"Start":"03:37.340 ","End":"03:40.100","Text":"We get b is 0 when a is 1,"},{"Start":"03:40.100 ","End":"03:41.375","Text":"and if you remember,"},{"Start":"03:41.375 ","End":"03:44.780","Text":"y was A sine x plus B cosine of x,"},{"Start":"03:44.780 ","End":"03:47.700","Text":"so 1 sine x plus 0 cosine x,"},{"Start":"03:47.700 ","End":"03:49.950","Text":"in short it\u0027s just sine x."},{"Start":"03:49.950 ","End":"03:53.655","Text":"The last step is just to combine."},{"Start":"03:53.655 ","End":"03:59.365","Text":"We take the homogeneous bit plus the private bit,"},{"Start":"03:59.365 ","End":"04:03.000","Text":"the general y is yh,"},{"Start":"04:03.000 ","End":"04:05.430","Text":"general solution of homogeneous,"},{"Start":"04:05.430 ","End":"04:07.275","Text":"plus the y particular."},{"Start":"04:07.275 ","End":"04:08.910","Text":"This bit is the homogeneous,"},{"Start":"04:08.910 ","End":"04:10.020","Text":"this bit is the particular,"},{"Start":"04:10.020 ","End":"04:14.050","Text":"that\u0027s the general solution, and we\u0027re done."}],"ID":7804},{"Watched":false,"Name":"Exercise 2","Duration":"3m 36s","ChapterTopicVideoID":7736,"CourseChapterTopicPlaylistID":4229,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"This differential equation is of the fourth order."},{"Start":"00:03.630 ","End":"00:10.080","Text":"It\u0027s still linear constant coefficients and it\u0027s not homogeneous."},{"Start":"00:10.080 ","End":"00:12.360","Text":"There\u0027s more than one way to handle that."},{"Start":"00:12.360 ","End":"00:16.050","Text":"We\u0027re going to use the method of undetermined coefficients."},{"Start":"00:16.050 ","End":"00:17.550","Text":"But in any event,"},{"Start":"00:17.550 ","End":"00:19.980","Text":"we want to solve the homogeneous first,"},{"Start":"00:19.980 ","End":"00:23.985","Text":"which as you know means just putting 0 on the right instead."},{"Start":"00:23.985 ","End":"00:25.920","Text":"Then to solve the homogeneous,"},{"Start":"00:25.920 ","End":"00:29.250","Text":"we start with the characteristic equation."},{"Start":"00:29.250 ","End":"00:32.925","Text":"Here it is, but it\u0027s of degree 4."},{"Start":"00:32.925 ","End":"00:34.530","Text":"How do we solve it?"},{"Start":"00:34.530 ","End":"00:36.905","Text":"One technique we can try,"},{"Start":"00:36.905 ","End":"00:39.170","Text":"since the leading coefficient is 1,"},{"Start":"00:39.170 ","End":"00:44.030","Text":"is to say that any integer solution has to divide 30."},{"Start":"00:44.030 ","End":"00:46.490","Text":"That\u0027s still gives a lot of possibilities."},{"Start":"00:46.490 ","End":"00:52.560","Text":"Actually 14 possibilities, I make it and we just try them one-by-one."},{"Start":"00:52.560 ","End":"00:54.530","Text":"I just noticed a typo."},{"Start":"00:54.530 ","End":"01:00.200","Text":"Well I forgot the plus or minus 316 possibilities now."},{"Start":"01:00.200 ","End":"01:03.620","Text":"Well, I\u0027m going to cut to the chase and just tell you after plugging all"},{"Start":"01:03.620 ","End":"01:07.190","Text":"of the main or we can stop once we\u0027ve reached four solutions."},{"Start":"01:07.190 ","End":"01:10.730","Text":"Once you\u0027ve reached, you\u0027ve done the plus or minus 1,2,3,"},{"Start":"01:10.730 ","End":"01:14.930","Text":"and 5 you already have found four different solutions and now you can"},{"Start":"01:14.930 ","End":"01:20.140","Text":"stop because the fourth degree polynomial can have up to four solutions."},{"Start":"01:20.140 ","End":"01:26.405","Text":"The general homogeneous is the linear combination of e to the kx,"},{"Start":"01:26.405 ","End":"01:29.435","Text":"where k is 1 minus 2,"},{"Start":"01:29.435 ","End":"01:32.110","Text":"3 and minus 5."},{"Start":"01:32.110 ","End":"01:36.335","Text":"The next step will be to find a particular solution."},{"Start":"01:36.335 ","End":"01:44.855","Text":"Now I want to remind you the right-hand side of the original OD was minus 28 e to the 2x."},{"Start":"01:44.855 ","End":"01:48.230","Text":"Now if we successively differentiate this,"},{"Start":"01:48.230 ","End":"01:51.110","Text":"we keep getting something e to the 2x."},{"Start":"01:51.110 ","End":"01:54.900","Text":"Really only the e to the 2x matters."},{"Start":"01:54.900 ","End":"02:02.195","Text":"Our general solution will be of the form some constant times e to the 2x."},{"Start":"02:02.195 ","End":"02:05.870","Text":"Now this has to satisfy the original OD,"},{"Start":"02:05.870 ","End":"02:09.760","Text":"so we need to differentiate it 4 times actually."},{"Start":"02:09.760 ","End":"02:14.125","Text":"Here they are each time it\u0027s multiplied by 2 from here."},{"Start":"02:14.125 ","End":"02:15.860","Text":"This is the fourth derivative."},{"Start":"02:15.860 ","End":"02:19.370","Text":"We more commonly write a 4 like that."},{"Start":"02:19.370 ","End":"02:24.385","Text":"Now we substitute in the original equation which was this."},{"Start":"02:24.385 ","End":"02:26.090","Text":"This is what we get, for example,"},{"Start":"02:26.090 ","End":"02:28.040","Text":"y to the fourth I copy from here."},{"Start":"02:28.040 ","End":"02:31.520","Text":"Then we have plus 3y triple prime."},{"Start":"02:31.520 ","End":"02:36.950","Text":"We have 3 times this is this and so on and so on."},{"Start":"02:36.950 ","End":"02:40.500","Text":"Here was the original right-hand side."},{"Start":"02:40.500 ","End":"02:44.530","Text":"Everything is in terms of e to the 2x."},{"Start":"02:44.530 ","End":"02:47.075","Text":"I just have to collect the terms."},{"Start":"02:47.075 ","End":"02:49.775","Text":"First, I take the e to 2x out."},{"Start":"02:49.775 ","End":"02:51.200","Text":"I can also take the A out,"},{"Start":"02:51.200 ","End":"02:53.155","Text":"just count how many A\u0027s there are."},{"Start":"02:53.155 ","End":"02:57.590","Text":"Computation shows this is minus 28A equals then"},{"Start":"02:57.590 ","End":"03:01.970","Text":"minus 28 because we compare the coefficients,"},{"Start":"03:01.970 ","End":"03:05.989","Text":"ignore the e to the 2x and that solves very conveniently,"},{"Start":"03:05.989 ","End":"03:07.630","Text":"obviously, A is 1."},{"Start":"03:07.630 ","End":"03:10.785","Text":"Remember our solution was e to the 2x."},{"Start":"03:10.785 ","End":"03:13.290","Text":"Here we have 1e to the 2x."},{"Start":"03:13.290 ","End":"03:14.730","Text":"I didn\u0027t write the 1."},{"Start":"03:14.730 ","End":"03:16.320","Text":"Now we have yp,"},{"Start":"03:16.320 ","End":"03:17.620","Text":"we also have yh,"},{"Start":"03:17.620 ","End":"03:21.545","Text":"so we combine them to get the general solution,"},{"Start":"03:21.545 ","End":"03:31.455","Text":"which is just all this part is the yh and this part is the yp."},{"Start":"03:31.455 ","End":"03:36.330","Text":"We add them together to get the general y. We\u0027re done."}],"ID":7805},{"Watched":false,"Name":"Exercise 3","Duration":"3m 36s","ChapterTopicVideoID":7734,"CourseChapterTopicPlaylistID":4229,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"Here we have another non-homogeneous equation."},{"Start":"00:03.585 ","End":"00:08.405","Text":"This time it\u0027s of the third-order linear constant coefficients,"},{"Start":"00:08.405 ","End":"00:13.185","Text":"and we\u0027re going to do it with the method of undetermined coefficients."},{"Start":"00:13.185 ","End":"00:14.490","Text":"But first of all,"},{"Start":"00:14.490 ","End":"00:18.120","Text":"we start by solving the homogeneous, which is this."},{"Start":"00:18.120 ","End":"00:23.625","Text":"Just put a 0 on the right and we know how to do this characteristic equation,"},{"Start":"00:23.625 ","End":"00:25.650","Text":"but it\u0027s a cubic equation."},{"Start":"00:25.650 ","End":"00:28.440","Text":"We don\u0027t know the general formula for solving it,"},{"Start":"00:28.440 ","End":"00:29.954","Text":"but we have a few tricks."},{"Start":"00:29.954 ","End":"00:34.710","Text":"1 of them is decomposition by groups because we see there\u0027s a 1 and 2,"},{"Start":"00:34.710 ","End":"00:36.510","Text":"1 and 2 on the opposite side."},{"Start":"00:36.510 ","End":"00:41.445","Text":"We take k minus 2 out of the brackets in both and then we combine these."},{"Start":"00:41.445 ","End":"00:44.670","Text":"Take k-2 out to get k^2-1."},{"Start":"00:44.670 ","End":"00:50.115","Text":"You can factorize k^2-1 as (k-1) (k+1)."},{"Start":"00:50.115 ","End":"00:52.009","Text":"Now we have a completely factorize,"},{"Start":"00:52.009 ","End":"00:53.930","Text":"so we know the 3 solutions."},{"Start":"00:53.930 ","End":"00:58.035","Text":"It will be 2,1, and -1."},{"Start":"00:58.035 ","End":"01:02.240","Text":"The homogeneous general solution is this."},{"Start":"01:02.240 ","End":"01:03.320","Text":"The 2 goes here."},{"Start":"01:03.320 ","End":"01:05.090","Text":"The 1 goes with the 1x,"},{"Start":"01:05.090 ","End":"01:08.420","Text":"which is just x-1x minus x."},{"Start":"01:08.420 ","End":"01:11.540","Text":"The next step will be to find a particular solution."},{"Start":"01:11.540 ","End":"01:14.330","Text":"Here again, is the original equation."},{"Start":"01:14.330 ","End":"01:17.610","Text":"Now we want to see which functions appear on the right"},{"Start":"01:17.610 ","End":"01:22.220","Text":"and to differentiate them repeatedly and see what we get."},{"Start":"01:22.220 ","End":"01:24.160","Text":"If we take the first bit, the 2x^3,"},{"Start":"01:24.160 ","End":"01:27.650","Text":"and keep differentiating it and ignoring constants,"},{"Start":"01:27.650 ","End":"01:30.365","Text":"we get x^3, x^2, x, and 1."},{"Start":"01:30.365 ","End":"01:33.460","Text":"The same thing if we differentiate any of these,"},{"Start":"01:33.460 ","End":"01:35.340","Text":"let\u0027s look at the same building blocks,"},{"Start":"01:35.340 ","End":"01:38.010","Text":"x^3, x^2, x, and 1."},{"Start":"01:38.010 ","End":"01:40.519","Text":"By the method of undetermined coefficients,"},{"Start":"01:40.519 ","End":"01:45.910","Text":"we\u0027re going to look for a particular solution of the form linear combination of these,"},{"Start":"01:45.910 ","End":"01:47.930","Text":"meaning a times this,"},{"Start":"01:47.930 ","End":"01:53.300","Text":"b times this plus c times this plus d and that\u0027s going to be our y particular."},{"Start":"01:53.300 ","End":"01:54.860","Text":"In order for it to be a solution,"},{"Start":"01:54.860 ","End":"01:56.960","Text":"it has to satisfy the original equation."},{"Start":"01:56.960 ","End":"02:00.335","Text":"Of course, if I plug this into here,"},{"Start":"02:00.335 ","End":"02:02.000","Text":"it should satisfy it."},{"Start":"02:02.000 ","End":"02:03.900","Text":"Now here\u0027s the first, second,"},{"Start":"02:03.900 ","End":"02:07.220","Text":"and third derivatives, pretty straightforward."},{"Start":"02:07.220 ","End":"02:12.810","Text":"Now the substitution for convenience of writing this down again."},{"Start":"02:12.920 ","End":"02:16.340","Text":"Here\u0027s what we get. Well, the right-hand side is here,"},{"Start":"02:16.340 ","End":"02:17.600","Text":"just the right-hand side."},{"Start":"02:17.600 ","End":"02:22.010","Text":"Now for the left, I just look at the y triple prime is 6A, that\u0027s here."},{"Start":"02:22.010 ","End":"02:28.345","Text":"Minus 2y double prime -2 and I copy y double prime and so on and so on."},{"Start":"02:28.345 ","End":"02:32.420","Text":"I rearrange the left-hand side to collect together x^3,"},{"Start":"02:32.420 ","End":"02:35.045","Text":"x^2 is x as in constants."},{"Start":"02:35.045 ","End":"02:38.524","Text":"I\u0027m going to just copy the right-hand side in a second."},{"Start":"02:38.524 ","End":"02:43.730","Text":"Here it is, but I put it in reverse order because this is in reverse order."},{"Start":"02:43.730 ","End":"02:48.005","Text":"Then it\u0027s easy to see that we want this to be 14,"},{"Start":"02:48.005 ","End":"02:50.915","Text":"this part to be minus 12, and so on."},{"Start":"02:50.915 ","End":"02:55.130","Text":"This way we get 4 equations and 4 unknowns, a, b, c,"},{"Start":"02:55.130 ","End":"02:59.750","Text":"d. I\u0027m giving you the solutions because we don\u0027t want to waste time solving this."},{"Start":"02:59.750 ","End":"03:03.845","Text":"That\u0027s not what we\u0027re here to study at the moment. That\u0027s that."},{"Start":"03:03.845 ","End":"03:08.540","Text":"Now, remember that the particular solution was this"},{"Start":"03:08.540 ","End":"03:13.610","Text":"times x^3 plus this times x^2 plus this times x plus this."},{"Start":"03:13.610 ","End":"03:16.820","Text":"The x^2 and the x coefficients are both zeros,"},{"Start":"03:16.820 ","End":"03:21.334","Text":"so we\u0027re just left with x^3 plus 4 for the particular solution."},{"Start":"03:21.334 ","End":"03:23.510","Text":"Now if I add this to the homogeneous,"},{"Start":"03:23.510 ","End":"03:24.965","Text":"that\u0027s my last step,"},{"Start":"03:24.965 ","End":"03:27.050","Text":"we get the general solution."},{"Start":"03:27.050 ","End":"03:30.635","Text":"The first bit is the homogeneous,"},{"Start":"03:30.635 ","End":"03:33.110","Text":"then this is the particular add them together"},{"Start":"03:33.110 ","End":"03:36.690","Text":"and that\u0027s the general solution. We\u0027re done."}],"ID":7806},{"Watched":false,"Name":"Exercise 4","Duration":"5m 14s","ChapterTopicVideoID":7738,"CourseChapterTopicPlaylistID":4229,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"Here we have another non homogeneous equation"},{"Start":"00:04.380 ","End":"00:07.545","Text":"which we\u0027ll solve using undetermined coefficients."},{"Start":"00:07.545 ","End":"00:10.455","Text":"As usual, we start with the homogeneous,"},{"Start":"00:10.455 ","End":"00:13.920","Text":"which is the same except on the right hand side I put 0."},{"Start":"00:13.920 ","End":"00:15.780","Text":"By the way, I want you to remember"},{"Start":"00:15.780 ","End":"00:18.090","Text":"this right-hand side because I\u0027m going to"},{"Start":"00:18.090 ","End":"00:20.610","Text":"use it later and this will disappear off screen."},{"Start":"00:20.610 ","End":"00:25.605","Text":"Back to the homogeneous we find the characteristic equation which is this."},{"Start":"00:25.605 ","End":"00:30.899","Text":"But it\u0027s cubic, so 1 way is to guess the roots."},{"Start":"00:30.899 ","End":"00:34.380","Text":"We use a system that we have whole number roots."},{"Start":"00:34.380 ","End":"00:37.770","Text":"They must be divisors of this 2."},{"Start":"00:37.770 ","End":"00:40.935","Text":"That gives us these possibilities."},{"Start":"00:40.935 ","End":"00:44.475","Text":"We just substitute each 1 and see what works."},{"Start":"00:44.475 ","End":"00:49.620","Text":"I\u0027ve done that and the only 2 of them work 1 and minus 2."},{"Start":"00:49.620 ","End":"00:54.290","Text":"For example, 1^3-3*1+2 is 1-3+2=0, and so on."},{"Start":"00:54.290 ","End":"00:56.270","Text":"If we substitute this, it also works,"},{"Start":"00:56.270 ","End":"00:59.685","Text":"but we need 3 roots because it\u0027s a cubic."},{"Start":"00:59.685 ","End":"01:03.000","Text":"Probably, or possibly 1 of these is"},{"Start":"01:03.000 ","End":"01:08.090","Text":"a double root and we can check that by seeing if it\u0027s a root of the derivative."},{"Start":"01:08.090 ","End":"01:11.460","Text":"Derivative of this is this and you can see that"},{"Start":"01:11.460 ","End":"01:15.285","Text":"1 is a root of this so 1 is a double root."},{"Start":"01:15.285 ","End":"01:17.370","Text":"That\u0027s a third root altogether,"},{"Start":"01:17.370 ","End":"01:22.665","Text":"we have 1,1 and minus 2."},{"Start":"01:22.665 ","End":"01:28.850","Text":"That means that the solution of the differential equation, the homogeneous,"},{"Start":"01:28.850 ","End":"01:30.320","Text":"will be as follows,"},{"Start":"01:30.320 ","End":"01:36.035","Text":"the coefficients of the exponent or 1,1 and minus 2."},{"Start":"01:36.035 ","End":"01:38.525","Text":"But because this was a double root,"},{"Start":"01:38.525 ","End":"01:40.820","Text":"then we get the extra x here."},{"Start":"01:40.820 ","End":"01:43.280","Text":"Remember the right-hand side is e^x."},{"Start":"01:43.280 ","End":"01:45.890","Text":"I\u0027m going to use this in part 2."},{"Start":"01:45.890 ","End":"01:50.230","Text":"We remember that the right-hand side was e^x."},{"Start":"01:50.230 ","End":"01:53.030","Text":"But e^x, when you keep differentiating,"},{"Start":"01:53.030 ","End":"01:56.555","Text":"it is just e^x, e^x."},{"Start":"01:56.555 ","End":"01:58.860","Text":"All we need is e^x."},{"Start":"01:58.860 ","End":"02:01.290","Text":"Using undetermined coefficients,"},{"Start":"02:01.290 ","End":"02:05.695","Text":"I look for a solution of the form Ae^x."},{"Start":"02:05.695 ","End":"02:09.610","Text":"Now if you recall in the tutorial there was an exception,"},{"Start":"02:09.610 ","End":"02:11.440","Text":"I call that the glitch."},{"Start":"02:11.440 ","End":"02:14.020","Text":"That\u0027s why I wrote the word initial here."},{"Start":"02:14.020 ","End":"02:15.430","Text":"That\u0027s just the first guess."},{"Start":"02:15.430 ","End":"02:17.140","Text":"We\u0027re going to have to correct it."},{"Start":"02:17.140 ","End":"02:24.780","Text":"The reason for that is that this e^x also appears in the homogeneous."},{"Start":"02:24.780 ","End":"02:26.890","Text":"When we have a duplication,"},{"Start":"02:26.890 ","End":"02:30.580","Text":"what we have to do is multiply by x"},{"Start":"02:30.580 ","End":"02:35.310","Text":"and we might have to do that more than once because of the clash,"},{"Start":"02:35.310 ","End":"02:39.930","Text":"we try next with xe^x."},{"Start":"02:39.930 ","End":"02:45.515","Text":"But we still get a clash because xe^x also appears."},{"Start":"02:45.515 ","End":"02:50.360","Text":"It could appear either in the homogeneous or elsewhere in the initial guess,"},{"Start":"02:50.360 ","End":"02:54.110","Text":"but there\u0027s only 1 term in this case so that doesn\u0027t apply."},{"Start":"02:54.110 ","End":"02:56.375","Text":"But if we have a clash either here or here,"},{"Start":"02:56.375 ","End":"02:59.135","Text":"then we just multiply by x."},{"Start":"02:59.135 ","End":"03:03.690","Text":"Next attempt will be ax^e^x."},{"Start":"03:04.090 ","End":"03:07.730","Text":"Now we\u0027re clear because it doesn\u0027t clash with"},{"Start":"03:07.730 ","End":"03:11.210","Text":"anything here and there\u0027s nothing else in the particular."},{"Start":"03:11.210 ","End":"03:18.125","Text":"We\u0027ll go with this and I\u0027ll emphasize it in red and I\u0027ve dropped the word initial now."},{"Start":"03:18.125 ","End":"03:22.610","Text":"This is what we\u0027re going for and what we need to do is"},{"Start":"03:22.610 ","End":"03:27.095","Text":"verify or make it so that it\u0027s a solution of the ODE."},{"Start":"03:27.095 ","End":"03:30.995","Text":"We\u0027ll need the first 3 derivatives."},{"Start":"03:30.995 ","End":"03:32.480","Text":"I\u0027ve just presented them,"},{"Start":"03:32.480 ","End":"03:34.610","Text":"the first, second, third derivatives."},{"Start":"03:34.610 ","End":"03:36.170","Text":"You can check the calculations,"},{"Start":"03:36.170 ","End":"03:42.120","Text":"it\u0027s pretty standard and I brought back the original equation."},{"Start":"03:42.120 ","End":"03:44.040","Text":"Now we have to substitute,"},{"Start":"03:44.040 ","End":"03:46.335","Text":"though we don\u0027t need this 1."},{"Start":"03:46.335 ","End":"03:48.769","Text":"We get the triple prime,"},{"Start":"03:48.769 ","End":"03:50.120","Text":"we just copy from here,"},{"Start":"03:50.120 ","End":"03:53.630","Text":"that\u0027s this, y prime I take from here,"},{"Start":"03:53.630 ","End":"03:55.535","Text":"and it was a minus 3."},{"Start":"03:55.535 ","End":"04:00.110","Text":"Lastly, plus 2 times what was here."},{"Start":"04:00.110 ","End":"04:04.985","Text":"This has to equal the e^x that was there originally."},{"Start":"04:04.985 ","End":"04:06.965","Text":"I need a bit more space here."},{"Start":"04:06.965 ","End":"04:14.690","Text":"Here I collected like terms and there\u0027s a little typo this should read 6A, sorry."},{"Start":"04:14.690 ","End":"04:21.970","Text":"Now if you notice A-3A+2A=0 and 6A-6A=0."},{"Start":"04:21.970 ","End":"04:31.095","Text":"Really, this equation just says that 6Ae^x=e^x as functions"},{"Start":"04:31.095 ","End":"04:34.950","Text":"and so we get that 6A=1 I mean this is just 1e^x so"},{"Start":"04:34.950 ","End":"04:41.380","Text":"these coefficients have to be equal and A=1_6."},{"Start":"04:42.350 ","End":"04:46.965","Text":"The y_p which was Ax^2e^x."},{"Start":"04:46.965 ","End":"04:49.800","Text":"There it is even here that was our y_p."},{"Start":"04:49.800 ","End":"04:54.615","Text":"A=1_6 so it\u0027s 1_6x^2e^x."},{"Start":"04:54.615 ","End":"04:57.810","Text":"Now that we have y_p and we already had y_h,"},{"Start":"04:57.810 ","End":"04:59.580","Text":"the general solution of the homogeneous,"},{"Start":"04:59.580 ","End":"05:02.640","Text":"we just have to combine them and here we are."},{"Start":"05:02.640 ","End":"05:07.235","Text":"The first 3 terms are solution to the homogeneous which we had above."},{"Start":"05:07.235 ","End":"05:10.585","Text":"The last term is the y_p,"},{"Start":"05:10.585 ","End":"05:15.110","Text":"and this is the general solution and we are done."}],"ID":7807},{"Watched":false,"Name":"Exercise 5","Duration":"6m 24s","ChapterTopicVideoID":7737,"CourseChapterTopicPlaylistID":4229,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"Here we have another one of these non-homogeneous differential equations,"},{"Start":"00:05.310 ","End":"00:09.510","Text":"which we\u0027re going to solve using the method of undetermined coefficients."},{"Start":"00:09.510 ","End":"00:10.680","Text":"But first of all,"},{"Start":"00:10.680 ","End":"00:12.810","Text":"we have to solve the homogeneous,"},{"Start":"00:12.810 ","End":"00:17.340","Text":"which is the same except that put 0 on the right-hand side."},{"Start":"00:17.340 ","End":"00:19.440","Text":"Just want to make a note to this,"},{"Start":"00:19.440 ","End":"00:26.010","Text":"we\u0027ll try and remember that we\u0027ll use it later when we go for the particular y in part 2."},{"Start":"00:26.010 ","End":"00:28.250","Text":"Anyway, the homogeneous we know how to solve,"},{"Start":"00:28.250 ","End":"00:30.440","Text":"we start off with the characteristic."},{"Start":"00:30.440 ","End":"00:32.765","Text":"This is a cubic equation."},{"Start":"00:32.765 ","End":"00:36.470","Text":"But here I think we can see that we"},{"Start":"00:36.470 ","End":"00:41.630","Text":"have something minus something with 1 minus 1, 1 minus 1."},{"Start":"00:41.630 ","End":"00:47.770","Text":"There\u0027s a pattern, we could take k minus 1 outside of each group and we get this."},{"Start":"00:47.770 ","End":"00:51.005","Text":"This is the factorization by groups method."},{"Start":"00:51.005 ","End":"00:55.735","Text":"From here, we can take k minus 1 out of the whole thing."},{"Start":"00:55.735 ","End":"00:57.090","Text":"This is what we get,"},{"Start":"00:57.090 ","End":"00:59.990","Text":"so one solution obviously is k=1,"},{"Start":"00:59.990 ","End":"01:01.625","Text":"but here\u0027s a quadratic."},{"Start":"01:01.625 ","End":"01:04.475","Text":"This is going to give us 2 complex solutions."},{"Start":"01:04.475 ","End":"01:10.580","Text":"This is just what I said about the k=1 and this one, k^2+1,"},{"Start":"01:10.580 ","End":"01:14.540","Text":"we can just solve by bringing the one to the other side and we have"},{"Start":"01:14.540 ","End":"01:18.665","Text":"the 2 square roots of minus 1 plus or minus i,"},{"Start":"01:18.665 ","End":"01:21.935","Text":"so this will be k_1, this a k_2, and k_3,"},{"Start":"01:21.935 ","End":"01:26.435","Text":"and that gives us the solution for the homogeneous."},{"Start":"01:26.435 ","End":"01:28.835","Text":"The first term comes from the k=1,"},{"Start":"01:28.835 ","End":"01:30.740","Text":"and that\u0027s fairly straightforward."},{"Start":"01:30.740 ","End":"01:34.085","Text":"Of course, don\u0027t write 1x, so we would just write x."},{"Start":"01:34.085 ","End":"01:38.270","Text":"These second and third terms come from the formula."},{"Start":"01:38.270 ","End":"01:41.600","Text":"When we have 2 complex conjugate solutions, in our case,"},{"Start":"01:41.600 ","End":"01:45.440","Text":"we can write it as 0 plus or minus 1i."},{"Start":"01:45.440 ","End":"01:50.840","Text":"Then the contribution is a^0x,"},{"Start":"01:50.840 ","End":"01:53.015","Text":"which of course is 1, I don\u0027t need it."},{"Start":"01:53.015 ","End":"01:55.600","Text":"Then b is 1,"},{"Start":"01:55.600 ","End":"02:01.670","Text":"so it\u0027s just cosine x plus sine x with constants in front here. That\u0027s that."},{"Start":"02:01.670 ","End":"02:03.410","Text":"Now that\u0027s the homogeneous."},{"Start":"02:03.410 ","End":"02:06.350","Text":"Now what about the particular solution?"},{"Start":"02:06.350 ","End":"02:10.783","Text":"If you recall, the right-hand side of the original equation was sine x,"},{"Start":"02:10.783 ","End":"02:14.165","Text":"and we use this to help us with y particular,"},{"Start":"02:14.165 ","End":"02:16.490","Text":"just keep differentiating, get"},{"Start":"02:16.490 ","End":"02:20.630","Text":"cosine x and after this it starts to repeat because we will get what?"},{"Start":"02:20.630 ","End":"02:24.380","Text":"Minus sine x but a constant is an important minus cosine x."},{"Start":"02:24.380 ","End":"02:25.820","Text":"If you differentiate one,"},{"Start":"02:25.820 ","End":"02:26.945","Text":"you get the other,"},{"Start":"02:26.945 ","End":"02:29.435","Text":"maybe with a plus or minus."},{"Start":"02:29.435 ","End":"02:35.450","Text":"Our initial guess for a particular solution will be"},{"Start":"02:35.450 ","End":"02:38.850","Text":"a times this plus b times this and A and"},{"Start":"02:38.850 ","End":"02:42.530","Text":"B are the undetermined constants and we want to determine them."},{"Start":"02:42.530 ","End":"02:46.100","Text":"But note that I wrote the word initial"},{"Start":"02:46.100 ","End":"02:50.329","Text":"here because remember there\u0027s the exceptional case,"},{"Start":"02:50.329 ","End":"02:53.585","Text":"what I call the glitch and the standard formula."},{"Start":"02:53.585 ","End":"02:58.385","Text":"Because this sine x appears also here in the"},{"Start":"02:58.385 ","End":"03:04.130","Text":"homogeneous and the cosine x also appears in the homogeneous,"},{"Start":"03:04.130 ","End":"03:07.265","Text":"so each of them will have to be multiplied by x,"},{"Start":"03:07.265 ","End":"03:11.420","Text":"and so on and keep multiplying by x until we get no conflict."},{"Start":"03:11.420 ","End":"03:15.940","Text":"Well, already we can see that x sine x and x cosine x are not going to be a conflict,"},{"Start":"03:15.940 ","End":"03:20.845","Text":"so I can multiply the whole thing in one go by x,"},{"Start":"03:20.845 ","End":"03:23.365","Text":"which I\u0027ve marked in red for emphasis."},{"Start":"03:23.365 ","End":"03:25.120","Text":"Now I can drop the word initial."},{"Start":"03:25.120 ","End":"03:27.790","Text":"This is our guess for y particular,"},{"Start":"03:27.790 ","End":"03:31.565","Text":"and we just have to determine the undetermined coefficients A and B."},{"Start":"03:31.565 ","End":"03:36.850","Text":"Now, this solution, I mean that\u0027s a satisfy the original ODE,"},{"Start":"03:36.850 ","End":"03:39.190","Text":"so we\u0027ll need its derivatives."},{"Start":"03:39.190 ","End":"03:41.050","Text":"I won\u0027t go into all the details."},{"Start":"03:41.050 ","End":"03:42.895","Text":"It\u0027s just straightforward in technical."},{"Start":"03:42.895 ","End":"03:45.010","Text":"For example, y_p\u0027,"},{"Start":"03:45.010 ","End":"03:46.150","Text":"I use the product rule."},{"Start":"03:46.150 ","End":"03:52.540","Text":"The derivative of x is 1 and then this one as is plus x as is and the derivative of this,"},{"Start":"03:52.540 ","End":"03:54.920","Text":"which is this, and so on."},{"Start":"03:54.920 ","End":"03:56.420","Text":"We keep differentiate again,"},{"Start":"03:56.420 ","End":"03:59.090","Text":"I get this and I tidy up and get this."},{"Start":"03:59.090 ","End":"04:02.410","Text":"Then I differentiate to get this and tidy up to get this."},{"Start":"04:02.410 ","End":"04:08.000","Text":"Now I\u0027ve got y and its first 3 derivatives for the y particular."},{"Start":"04:08.000 ","End":"04:12.290","Text":"Now I\u0027ve brought back the original equation and we\u0027re going to substitute y_p,"},{"Start":"04:12.290 ","End":"04:14.360","Text":"y_p\u0027, double prime,"},{"Start":"04:14.360 ","End":"04:15.640","Text":"triple prime in here."},{"Start":"04:15.640 ","End":"04:20.385","Text":"If you refer back to the previous page, this is y\u0027\u0027\u0027."},{"Start":"04:20.385 ","End":"04:22.155","Text":"Then we need minus y\u0027\u0027,"},{"Start":"04:22.155 ","End":"04:26.235","Text":"y\u0027\u0027 was this plus y\u0027,"},{"Start":"04:26.235 ","End":"04:33.240","Text":"I copied was this minus y was this and right-hand side just copied."},{"Start":"04:33.240 ","End":"04:37.265","Text":"Then I expand and collect together 4 types of term."},{"Start":"04:37.265 ","End":"04:39.490","Text":"I collected separately the sine x,"},{"Start":"04:39.490 ","End":"04:42.020","Text":"the cosine x, the x cosine x,"},{"Start":"04:42.020 ","End":"04:43.505","Text":"and the x sine x."},{"Start":"04:43.505 ","End":"04:45.650","Text":"These turn out to be the coefficients."},{"Start":"04:45.650 ","End":"04:47.495","Text":"This is equal to sine x,"},{"Start":"04:47.495 ","End":"04:50.070","Text":"which I can write as 1 sine x."},{"Start":"04:50.070 ","End":"04:54.200","Text":"Reason I\u0027m doing that is going to say that this coefficient has got to"},{"Start":"04:54.200 ","End":"04:59.105","Text":"be 1 it goes with the sine x and the other 3 have to be 0."},{"Start":"04:59.105 ","End":"05:04.760","Text":"Actually, we\u0027re in luck because this thing cancels out to be 0."},{"Start":"05:04.760 ","End":"05:11.590","Text":"I can still together cross this out and B plus A minus B minus A, that\u0027s also 0."},{"Start":"05:11.590 ","End":"05:17.315","Text":"I will get 2 equations that this is 1 and this is 0."},{"Start":"05:17.315 ","End":"05:22.580","Text":"Just do a final collecting minus 3A plus A is minus 2A plus"},{"Start":"05:22.580 ","End":"05:28.200","Text":"2B and this was minus 2B minus 2A."},{"Start":"05:28.200 ","End":"05:31.370","Text":"Again, it equals 1 sine x."},{"Start":"05:31.370 ","End":"05:36.395","Text":"As I said, that this one has to equal 1 and this one has to be 0."},{"Start":"05:36.395 ","End":"05:38.660","Text":"This is fairly straightforward,"},{"Start":"05:38.660 ","End":"05:41.540","Text":"2 equations in 2 unknowns,"},{"Start":"05:41.540 ","End":"05:43.040","Text":"and I\u0027ll just give you the solutions,"},{"Start":"05:43.040 ","End":"05:45.410","Text":"we won\u0027t waste time on this."},{"Start":"05:45.410 ","End":"05:48.230","Text":"Just to remind you because it\u0027s off the board here,"},{"Start":"05:48.230 ","End":"05:50.780","Text":"that y_p was what?"},{"Start":"05:50.780 ","End":"05:59.040","Text":"X times A sine x plus B cosine x and here it is."},{"Start":"05:59.040 ","End":"06:03.155","Text":"I just put A is minus 1/4 and B is 1/4 and that\u0027s that."},{"Start":"06:03.155 ","End":"06:05.830","Text":"Now we have the homogeneous and the particular,"},{"Start":"06:05.830 ","End":"06:08.760","Text":"and so we can get the general solution,"},{"Start":"06:08.760 ","End":"06:11.725","Text":"should have said that this is part 3."},{"Start":"06:11.725 ","End":"06:15.740","Text":"General solution is to take the homogeneous,"},{"Start":"06:15.740 ","End":"06:20.520","Text":"which was this plus the particular, which is this."},{"Start":"06:20.520 ","End":"06:25.270","Text":"That\u0027s our general solution and we\u0027re done."}],"ID":7808}],"Thumbnail":null,"ID":4229},{"Name":"The Wronskian and its Uses","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"3m 30s","ChapterTopicVideoID":7732,"CourseChapterTopicPlaylistID":4230,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/7732.jpeg","UploadDate":"2018-05-16T02:47:39.4200000","DurationForVideoObject":"PT3M30S","Description":null,"MetaTitle":"Exercise 1 - The Wronskian and its Uses: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on The Wronskian and its Uses practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/ordinary-differential-equations/n_th-order-linear-equations/the-wronskian-and-its-uses/vid7809","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"In this exercise, we\u0027re given three functions,"},{"Start":"00:03.285 ","End":"00:04.740","Text":"y_1, y_2, y_3,"},{"Start":"00:04.740 ","End":"00:06.710","Text":"which are x, x^2,"},{"Start":"00:06.710 ","End":"00:10.005","Text":"and x^3, and the question is,"},{"Start":"00:10.005 ","End":"00:16.745","Text":"is it possible that these are three solutions of our usual ODE,"},{"Start":"00:16.745 ","End":"00:18.705","Text":"this case a 3rd order,"},{"Start":"00:18.705 ","End":"00:25.185","Text":"linear, homogeneous, and with continuous coefficients on some interval,"},{"Start":"00:25.185 ","End":"00:29.430","Text":"in this case, 0 to Pi inclusive?"},{"Start":"00:29.430 ","End":"00:31.350","Text":"For the solution,"},{"Start":"00:31.350 ","End":"00:33.180","Text":"we going to use the Wronskian."},{"Start":"00:33.180 ","End":"00:37.370","Text":"Wronskian pretty much is our main tool in all of this."},{"Start":"00:37.370 ","End":"00:38.959","Text":"We take the three functions,"},{"Start":"00:38.959 ","End":"00:40.325","Text":"x, x^2, x^3,"},{"Start":"00:40.325 ","End":"00:45.470","Text":"and then we want the derivatives on the next row and then in the second derivatives."},{"Start":"00:45.470 ","End":"00:47.015","Text":"Here\u0027s what we get."},{"Start":"00:47.015 ","End":"00:49.850","Text":"Trivial to check these."},{"Start":"00:49.850 ","End":"00:51.620","Text":"Switching to a new page."},{"Start":"00:51.620 ","End":"00:52.910","Text":"Now the 1st row,"},{"Start":"00:52.910 ","End":"00:54.350","Text":"everything is divisible by x,"},{"Start":"00:54.350 ","End":"00:56.525","Text":"so I can take x outside."},{"Start":"00:56.525 ","End":"00:59.215","Text":"I hope you remember your determinants."},{"Start":"00:59.215 ","End":"01:01.009","Text":"That\u0027s one of the properties."},{"Start":"01:01.009 ","End":"01:06.440","Text":"If you take any one of the rows or columns and take a factor out,"},{"Start":"01:06.440 ","End":"01:08.555","Text":"it comes out in front of the determinant."},{"Start":"01:08.555 ","End":"01:11.155","Text":"Now look, we have a 1 here and a 1 here."},{"Start":"01:11.155 ","End":"01:15.085","Text":"What I\u0027m going to do is subtract the first two rows."},{"Start":"01:15.085 ","End":"01:18.530","Text":"Some people write row operations with the arrow this way,"},{"Start":"01:18.530 ","End":"01:21.230","Text":"and sometimes the arrow is this way."},{"Start":"01:21.230 ","End":"01:26.050","Text":"This says that, replace R_2 by R_2 minus R_1."},{"Start":"01:26.050 ","End":"01:27.680","Text":"Or the other way of looking at it is,"},{"Start":"01:27.680 ","End":"01:31.400","Text":"take R_2 minus R_1 and put it into R_2."},{"Start":"01:31.400 ","End":"01:33.610","Text":"R_2 is second row, and so on."},{"Start":"01:33.610 ","End":"01:38.460","Text":"Basically what I mean is subtract the first row from the second row."},{"Start":"01:38.460 ","End":"01:41.310","Text":"The first and last one are going to stay the same,"},{"Start":"01:41.310 ","End":"01:43.680","Text":"and here I\u0027m going to do a subtraction,"},{"Start":"01:43.680 ","End":"01:46.320","Text":"1 minus 1 is 0,"},{"Start":"01:46.320 ","End":"01:49.845","Text":"2x minus x is x,"},{"Start":"01:49.845 ","End":"01:53.685","Text":"3x^2 minus x^2 is 2x^2."},{"Start":"01:53.685 ","End":"01:58.650","Text":"Now we\u0027ve got a column with the two zeros in it,"},{"Start":"01:58.650 ","End":"02:04.190","Text":"so what we can do is to expand along the 1st column,"},{"Start":"02:04.190 ","End":"02:07.820","Text":"1 times the determinant of what\u0027s left,"},{"Start":"02:07.820 ","End":"02:09.275","Text":"everything else is 0,"},{"Start":"02:09.275 ","End":"02:12.640","Text":"so basically what we get is x times 1,"},{"Start":"02:12.640 ","End":"02:17.780","Text":"I didn\u0027t write the times 1 times the determinant of this bit here."},{"Start":"02:17.780 ","End":"02:19.669","Text":"Probably should have written the 1 here."},{"Start":"02:19.669 ","End":"02:21.140","Text":"Anyway, I\u0027ll circle it here."},{"Start":"02:21.140 ","End":"02:24.560","Text":"Okay. Yeah. So the determinant is this times this,"},{"Start":"02:24.560 ","End":"02:27.075","Text":"minus this times this."},{"Start":"02:27.075 ","End":"02:31.195","Text":"We get 6x^3 minus 4x^3, which is 2x^3."},{"Start":"02:31.195 ","End":"02:37.925","Text":"I want to remind you that the interval we\u0027re talking about was 0,"},{"Start":"02:37.925 ","End":"02:39.834","Text":"Pi, including the 0."},{"Start":"02:39.834 ","End":"02:42.290","Text":"The reason I\u0027m emphasizing this is, remember,"},{"Start":"02:42.290 ","End":"02:46.245","Text":"there\u0027s what I call the all or nothing proposition,"},{"Start":"02:46.245 ","End":"02:52.310","Text":"that the Wronskian of solutions is either 0 everywhere or 0 nowhere."},{"Start":"02:52.310 ","End":"02:54.860","Text":"It can\u0027t be sometimes 0 and sometimes non-zero."},{"Start":"02:54.860 ","End":"02:57.695","Text":"But if you look at this closely on this interval,"},{"Start":"02:57.695 ","End":"03:04.820","Text":"then w is 0 at exactly one point where x is 0 or non-zero elsewhere,"},{"Start":"03:04.820 ","End":"03:06.635","Text":"so we have a mix and match."},{"Start":"03:06.635 ","End":"03:08.870","Text":"They don\u0027t have all one-off thing."},{"Start":"03:08.870 ","End":"03:12.185","Text":"Of course, on a different interval if it\u0027d be, and I don\u0027t know,"},{"Start":"03:12.185 ","End":"03:14.840","Text":"x bigger than 0,"},{"Start":"03:14.840 ","End":"03:18.545","Text":"not including the 0 would be fine, but zero\u0027s included."},{"Start":"03:18.545 ","End":"03:20.335","Text":"As things stand,"},{"Start":"03:20.335 ","End":"03:22.230","Text":"these y_1, y_2,"},{"Start":"03:22.230 ","End":"03:25.155","Text":"y_3 as given can\u0027t be solutions of"},{"Start":"03:25.155 ","End":"03:31.770","Text":"the ODE because they violate the proposition of the all or nothing."}],"ID":7809},{"Watched":false,"Name":"Exercise 2","Duration":"4m 41s","ChapterTopicVideoID":7733,"CourseChapterTopicPlaylistID":4230,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we\u0027re given 3 functions,"},{"Start":"00:03.480 ","End":"00:06.000","Text":"y_1 is this, y_2 is this,"},{"Start":"00:06.000 ","End":"00:07.650","Text":"and y_3 is this."},{"Start":"00:07.650 ","End":"00:11.475","Text":"First step, just compute the Wronskian of the 3."},{"Start":"00:11.475 ","End":"00:15.270","Text":"Part b, determine whether or not they\u0027re linearly"},{"Start":"00:15.270 ","End":"00:20.160","Text":"independent on the whole number line minus infinity to infinity."},{"Start":"00:20.160 ","End":"00:25.230","Text":"Then we\u0027re going to answer b again after we note that these are"},{"Start":"00:25.230 ","End":"00:30.540","Text":"actually 3 solutions to this very simple differential equation."},{"Start":"00:30.540 ","End":"00:32.040","Text":"Let\u0027s start with a,"},{"Start":"00:32.040 ","End":"00:35.475","Text":"the Wronskian is the determinant of a 3 by 3."},{"Start":"00:35.475 ","End":"00:41.310","Text":"First row, I just copy the functions and then here I differentiate, differentiate again."},{"Start":"00:41.310 ","End":"00:44.135","Text":"Of course I\u0027ve just indicated that that\u0027s what I\u0027m going to do."},{"Start":"00:44.135 ","End":"00:48.560","Text":"Let\u0027s now do the actual differentiation on second derivative."},{"Start":"00:48.560 ","End":"00:52.215","Text":"We take this differentiate it minus 1, differentiate it again."},{"Start":"00:52.215 ","End":"00:57.650","Text":"Here derivative, derivative again and here the derivative and the derivative again."},{"Start":"00:57.650 ","End":"01:01.345","Text":"Notice that the whole last row is 0,"},{"Start":"01:01.345 ","End":"01:04.070","Text":"and when that happens, the whole determinant is 0,"},{"Start":"01:04.070 ","End":"01:06.470","Text":"so our Wronskian is 0."},{"Start":"01:06.470 ","End":"01:09.125","Text":"Now in part b,"},{"Start":"01:09.125 ","End":"01:11.480","Text":"these are just 3 functions."},{"Start":"01:11.480 ","End":"01:14.540","Text":"We\u0027re not talking about solutions of a differential equation yet,"},{"Start":"01:14.540 ","End":"01:18.875","Text":"it\u0027ll come in part c. We only have a one way test."},{"Start":"01:18.875 ","End":"01:21.410","Text":"We know if the Wronskian is non-zero,"},{"Start":"01:21.410 ","End":"01:22.715","Text":"then they are independent."},{"Start":"01:22.715 ","End":"01:25.955","Text":"When we get 0, we just don\u0027t know."},{"Start":"01:25.955 ","End":"01:27.800","Text":"We can\u0027t use the Wronskian,"},{"Start":"01:27.800 ","End":"01:30.005","Text":"however, we can check directly."},{"Start":"01:30.005 ","End":"01:35.870","Text":"New page, I\u0027ll just copy the 3 functions and proceed as usual."},{"Start":"01:35.870 ","End":"01:42.350","Text":"We want to know if we can find 3 constants that are not all 0 to make this work."},{"Start":"01:42.350 ","End":"01:44.660","Text":"If we can, then they\u0027re dependent."},{"Start":"01:44.660 ","End":"01:46.220","Text":"If all we can find are 0,"},{"Start":"01:46.220 ","End":"01:48.665","Text":"0, 0, then they\u0027re independent."},{"Start":"01:48.665 ","End":"01:52.070","Text":"Multiply out and collect to see if it\u0027s just constants."},{"Start":"01:52.070 ","End":"01:53.825","Text":"We have 4c_1,"},{"Start":"01:53.825 ","End":"01:56.150","Text":"4c_2, 20c_3,"},{"Start":"01:56.150 ","End":"01:59.440","Text":"and with the x\u0027s we have minus c_1,"},{"Start":"01:59.440 ","End":"02:03.975","Text":"plus c_2 and plus c_3."},{"Start":"02:03.975 ","End":"02:07.160","Text":"This was not an equation in x or anything."},{"Start":"02:07.160 ","End":"02:10.190","Text":"It\u0027s an identity sometimes written with 3 lines."},{"Start":"02:10.190 ","End":"02:13.250","Text":"This is a linear expression,"},{"Start":"02:13.250 ","End":"02:16.860","Text":"something times x plus something and it\u0027s going to be always 0."},{"Start":"02:16.860 ","End":"02:21.815","Text":"The coefficient of x has to be 0 and the 3 coefficient has to be 0."},{"Start":"02:21.815 ","End":"02:26.205","Text":"That gives us 2 equations and 3 unknowns."},{"Start":"02:26.205 ","End":"02:31.720","Text":"When that happens, we usually can get an infinite number of solutions."},{"Start":"02:31.720 ","End":"02:33.935","Text":"Anyway, let\u0027s tidy up a bit."},{"Start":"02:33.935 ","End":"02:38.600","Text":"This first equation can be divided by 4 and"},{"Start":"02:38.600 ","End":"02:43.310","Text":"bringing it to what is called echelon form but never mind if you forgot about that."},{"Start":"02:43.310 ","End":"02:50.820","Text":"What I did here was we just add the first equation to the second equation,"},{"Start":"02:50.820 ","End":"02:53.205","Text":"this one as is and if I add the 2,"},{"Start":"02:53.205 ","End":"02:54.990","Text":"c_1 and minus c_1 is 0,"},{"Start":"02:54.990 ","End":"02:56.640","Text":"c_2 and c_2 is 2c_2,"},{"Start":"02:56.640 ","End":"02:58.350","Text":"5 and 1 is 6."},{"Start":"02:58.350 ","End":"03:00.845","Text":"This is what we have now."},{"Start":"03:00.845 ","End":"03:07.100","Text":"The last one could really be re-written as c_2 plus 3c_3 is 0,"},{"Start":"03:07.100 ","End":"03:12.505","Text":"just divide by 2 but notice that c_3 could be anything."},{"Start":"03:12.505 ","End":"03:18.465","Text":"Let\u0027s c_3 be anything and then I have 2 equations and 2 unknowns, c_1 and _2."},{"Start":"03:18.465 ","End":"03:22.830","Text":"Let c_3 be something not 0."},{"Start":"03:22.830 ","End":"03:27.465","Text":"A good example of not 0 is 1 so let us c_3 be 1."},{"Start":"03:27.465 ","End":"03:29.430","Text":"Once I\u0027ve let c_3 be 1,"},{"Start":"03:29.430 ","End":"03:32.625","Text":"the last equation tells me that c_2 plus 3 is 0,"},{"Start":"03:32.625 ","End":"03:39.015","Text":"so c_2 is minus 3 and once I have c_3 and c_2,"},{"Start":"03:39.015 ","End":"03:40.620","Text":"I plug them in here,"},{"Start":"03:40.620 ","End":"03:44.995","Text":"anyway c_1 comes out minus 2."},{"Start":"03:44.995 ","End":"03:47.285","Text":"I\u0027m just doing this to show that there is a solution."},{"Start":"03:47.285 ","End":"03:49.490","Text":"But even if 1 of them is non-zero,"},{"Start":"03:49.490 ","End":"03:52.380","Text":"that makes them already linearly dependent or linearly,"},{"Start":"03:52.380 ","End":"03:54.260","Text":"at least 1 of them to be non-zero."},{"Start":"03:54.260 ","End":"03:58.940","Text":"In fact, they\u0027re all 3 non-zero and then I\u0027ll just plug these back in and of course,"},{"Start":"03:58.940 ","End":"04:02.750","Text":"they\u0027re in backwards order minus 2 times the first function,"},{"Start":"04:02.750 ","End":"04:04.430","Text":"minus 3 times the second,"},{"Start":"04:04.430 ","End":"04:09.430","Text":"1 times the last is 0 proving that they are linearly dependent."},{"Start":"04:09.430 ","End":"04:11.900","Text":"That was a fair amount of work."},{"Start":"04:11.900 ","End":"04:14.495","Text":"But in part c,"},{"Start":"04:14.495 ","End":"04:17.255","Text":"they\u0027re not just any all 3 functions,"},{"Start":"04:17.255 ","End":"04:22.639","Text":"when we know that they are solutions to the type of ODE as above,"},{"Start":"04:22.639 ","End":"04:26.060","Text":"then we can use the results of the Wronskian."},{"Start":"04:26.060 ","End":"04:31.040","Text":"As you recall, the Wronskian came out 0 and for solutions,"},{"Start":"04:31.040 ","End":"04:33.140","Text":"we know that this means that they\u0027re linearly"},{"Start":"04:33.140 ","End":"04:37.655","Text":"dependent as opposed to just 3 plain functions."},{"Start":"04:37.655 ","End":"04:41.040","Text":"That answers part c and we\u0027re done."}],"ID":7810},{"Watched":false,"Name":"Exercise 3","Duration":"6m 28s","ChapterTopicVideoID":7731,"CourseChapterTopicPlaylistID":4230,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"This exercise isn\u0027t really an exercise you\u0027d get on an exam or anything,"},{"Start":"00:04.950 ","End":"00:10.065","Text":"it\u0027s just a theory as an example and disguised as an exercise."},{"Start":"00:10.065 ","End":"00:14.340","Text":"It\u0027s a generalization of something similar we had with 2 functions,"},{"Start":"00:14.340 ","End":"00:18.285","Text":"only this time it\u0027s 3 functions and it can be generalized to n functions."},{"Start":"00:18.285 ","End":"00:23.070","Text":"Let\u0027s assume that each of these is 3 times continuously differentiable,"},{"Start":"00:23.070 ","End":"00:28.395","Text":"meaning that they have continuous derivatives up to third order on an interval."},{"Start":"00:28.395 ","End":"00:32.880","Text":"That their Wronskian is non-zero and all the interval."},{"Start":"00:32.880 ","End":"00:36.600","Text":"We have to prove that there is a differential equation of"},{"Start":"00:36.600 ","End":"00:40.410","Text":"the kind we\u0027re familiar with linear, third-order,"},{"Start":"00:40.410 ","End":"00:43.720","Text":"homogeneous, and continuous coefficients,"},{"Start":"00:43.720 ","End":"00:46.950","Text":"in such a way that such an equation that y_1,"},{"Start":"00:46.950 ","End":"00:49.665","Text":"y_2, and y_3 are 3 of it\u0027s solutions."},{"Start":"00:49.665 ","End":"00:51.390","Text":"It\u0027s a reverse problem,"},{"Start":"00:51.390 ","End":"00:52.730","Text":"where usually we\u0027re given an equation,"},{"Start":"00:52.730 ","End":"00:54.440","Text":"we want to find solutions."},{"Start":"00:54.440 ","End":"00:56.359","Text":"Here we start with the opposite."},{"Start":"00:56.359 ","End":"00:58.809","Text":"These are going to be a fundamental set of solutions,"},{"Start":"00:58.809 ","End":"01:05.225","Text":"b is an example of a where we\u0027re given specifically that y_1 is x,"},{"Start":"01:05.225 ","End":"01:08.135","Text":"y_2 is x^2, y_3 is x^3,"},{"Start":"01:08.135 ","End":"01:12.890","Text":"and we\u0027re looking at the interval where x is bigger than 0."},{"Start":"01:12.890 ","End":"01:15.560","Text":"But let\u0027s start with part a and we\u0027re going to"},{"Start":"01:15.560 ","End":"01:18.815","Text":"use the same trick we used with 2 functions."},{"Start":"01:18.815 ","End":"01:24.005","Text":"The trick is to consider the following equation doesn\u0027t even quite look like an equation."},{"Start":"01:24.005 ","End":"01:25.850","Text":"But if you think about it,"},{"Start":"01:25.850 ","End":"01:30.905","Text":"all these 3 rows are just actual functions that are given."},{"Start":"01:30.905 ","End":"01:33.920","Text":"The top row remains as variables y,"},{"Start":"01:33.920 ","End":"01:35.930","Text":"y\u0027, y\u0027\u0027, y\u0027\u0027\u0027,"},{"Start":"01:35.930 ","End":"01:38.870","Text":"so when we expand this 4 by 4 determinant,"},{"Start":"01:38.870 ","End":"01:42.250","Text":"we\u0027re going get a differential equation of order 3."},{"Start":"01:42.250 ","End":"01:45.320","Text":"Let\u0027s start with processing this determinant"},{"Start":"01:45.320 ","End":"01:48.275","Text":"using our knowledge of determinants from linear algebra."},{"Start":"01:48.275 ","End":"01:49.730","Text":"But before we start expanding,"},{"Start":"01:49.730 ","End":"01:55.320","Text":"let\u0027s just note that we actually have that y_1,"},{"Start":"01:55.320 ","End":"01:58.215","Text":"y_2, and y_3 are all solutions,"},{"Start":"01:58.215 ","End":"02:00.780","Text":"because if you replace the top row,"},{"Start":"02:00.780 ","End":"02:03.015","Text":"y by y_1,"},{"Start":"02:03.015 ","End":"02:06.545","Text":"you\u0027ll get the first and second row similar, the same,"},{"Start":"02:06.545 ","End":"02:09.020","Text":"and therefore the determinant is going to be 0,"},{"Start":"02:09.020 ","End":"02:12.020","Text":"and if you replace them by y_2 or y_3,"},{"Start":"02:12.020 ","End":"02:16.220","Text":"similarly, we\u0027ll have the first row the same as the third or the fourth."},{"Start":"02:16.220 ","End":"02:19.505","Text":"All these 3 are in fact solutions."},{"Start":"02:19.505 ","End":"02:24.470","Text":"Now we just have to expand the determinant and then bring it into standard form."},{"Start":"02:24.470 ","End":"02:29.060","Text":"What we do is we expand along the first row."},{"Start":"02:29.060 ","End":"02:33.185","Text":"I\u0027m not going to put every detail in because just the idea."},{"Start":"02:33.185 ","End":"02:39.610","Text":"What happens is that we take y and then we multiply it by a 3 by 3 determinant,"},{"Start":"02:39.610 ","End":"02:44.375","Text":"which is what happens when you delete the first row and column."},{"Start":"02:44.375 ","End":"02:46.610","Text":"Then for y\u0027, we get a similar thing."},{"Start":"02:46.610 ","End":"02:47.900","Text":"We erase this and this,"},{"Start":"02:47.900 ","End":"02:50.000","Text":"and we get a 3 by 3 determinant."},{"Start":"02:50.000 ","End":"02:52.625","Text":"Now these 3 by 3 determinants,"},{"Start":"02:52.625 ","End":"02:54.935","Text":"I\u0027ll just give them names."},{"Start":"02:54.935 ","End":"02:57.245","Text":"The first one I\u0027ll call triangle,"},{"Start":"02:57.245 ","End":"02:59.120","Text":"the second one square,"},{"Start":"02:59.120 ","End":"03:02.045","Text":"and the third one parallelogram."},{"Start":"03:02.045 ","End":"03:09.210","Text":"The last one, we will actually expand and what we get is y\u0027\u0027\u0027 and it\u0027s minus,"},{"Start":"03:09.210 ","End":"03:12.380","Text":"remember it\u0027s alternation of sign plus, minus, plus,"},{"Start":"03:12.380 ","End":"03:16.765","Text":"minus y\u0027\u0027\u0027 times this determinant."},{"Start":"03:16.765 ","End":"03:20.075","Text":"Now this last determinant is actually the Wronskian."},{"Start":"03:20.075 ","End":"03:23.690","Text":"If you just change rows and columns and write this as the first row,"},{"Start":"03:23.690 ","End":"03:24.980","Text":"the second row and the third row,"},{"Start":"03:24.980 ","End":"03:28.400","Text":"you\u0027ll see that this is the Wronskian and determinants"},{"Start":"03:28.400 ","End":"03:32.405","Text":"don\u0027t change if you switch rows and columns."},{"Start":"03:32.405 ","End":"03:34.480","Text":"Then I\u0027ll get some more space."},{"Start":"03:34.480 ","End":"03:36.410","Text":"What I wrote here is just what I said,"},{"Start":"03:36.410 ","End":"03:40.790","Text":"that this last one is the Wronskian,"},{"Start":"03:40.790 ","End":"03:43.775","Text":"which we\u0027re told is non-zero."},{"Start":"03:43.775 ","End":"03:48.350","Text":"That\u0027s important because then we can divide by it and do a couple of other things."},{"Start":"03:48.350 ","End":"03:49.850","Text":"First of all, change the order."},{"Start":"03:49.850 ","End":"03:53.765","Text":"We have the derivatives in this order,"},{"Start":"03:53.765 ","End":"03:55.220","Text":"and it\u0027s going to be minus,"},{"Start":"03:55.220 ","End":"03:56.270","Text":"plus, minus plus,"},{"Start":"03:56.270 ","End":"03:58.340","Text":"so we just multiply everything by minus 1,"},{"Start":"03:58.340 ","End":"04:00.710","Text":"and we end up with this."},{"Start":"04:00.710 ","End":"04:02.480","Text":"We\u0027ve answered the question,"},{"Start":"04:02.480 ","End":"04:06.230","Text":"all we have to do is recognize that this will"},{"Start":"04:06.230 ","End":"04:12.349","Text":"minus with this is going to be p. This over this will be q"},{"Start":"04:12.349 ","End":"04:17.090","Text":"and minus this over this will be r. We can actually say"},{"Start":"04:17.090 ","End":"04:24.605","Text":"that y\u0027\u0027\u0027 plus p y\u0027\u0027 plus q,"},{"Start":"04:24.605 ","End":"04:28.775","Text":"y\u0027 plus r equals 0."},{"Start":"04:28.775 ","End":"04:30.575","Text":"That basically proves it,"},{"Start":"04:30.575 ","End":"04:34.055","Text":"even though we haven\u0027t got an exact formula here."},{"Start":"04:34.055 ","End":"04:35.985","Text":"But they\u0027re still functions,"},{"Start":"04:35.985 ","End":"04:38.970","Text":"and they\u0027re going to be continuous because"},{"Start":"04:38.970 ","End":"04:42.650","Text":"they\u0027re products and sums of continuous functions."},{"Start":"04:42.650 ","End":"04:46.110","Text":"Let\u0027s move on to Part b,"},{"Start":"04:46.110 ","End":"04:48.215","Text":"and I\u0027ll do it on a fresh page."},{"Start":"04:48.215 ","End":"04:49.685","Text":"Now this is what we get."},{"Start":"04:49.685 ","End":"04:53.330","Text":"The first row is y_1,"},{"Start":"04:53.330 ","End":"04:57.620","Text":"together with its derivative and the derivative and the derivative again,"},{"Start":"04:57.620 ","End":"05:02.645","Text":"then y_2, which is x^2 differentiate, differentiate, differentiate."},{"Start":"05:02.645 ","End":"05:04.500","Text":"Then y_3, which is x^3,"},{"Start":"05:04.500 ","End":"05:08.030","Text":"differentiate that, differentiate that, differentiate that."},{"Start":"05:08.030 ","End":"05:10.610","Text":"This time we\u0027re going to actually have to do the computations."},{"Start":"05:10.610 ","End":"05:12.140","Text":"I can\u0027t just write,"},{"Start":"05:12.140 ","End":"05:15.395","Text":"triangle, square, and parallelogram,"},{"Start":"05:15.395 ","End":"05:19.520","Text":"we\u0027re actually going to have to compute 3 determinants,"},{"Start":"05:19.520 ","End":"05:21.060","Text":"each of them 3 by 3."},{"Start":"05:21.060 ","End":"05:26.750","Text":"Now I haven\u0027t included all the tedious boring computations."},{"Start":"05:26.750 ","End":"05:29.390","Text":"The last one, if you figure it,"},{"Start":"05:29.390 ","End":"05:33.545","Text":"is the w comes out to be 2x^5,"},{"Start":"05:33.545 ","End":"05:35.420","Text":"which is not 0."},{"Start":"05:35.420 ","End":"05:39.890","Text":"The reason it\u0027s not 0 is our interval is x bigger than 0."},{"Start":"05:39.890 ","End":"05:41.675","Text":"Anyway, we end up with this,"},{"Start":"05:41.675 ","End":"05:45.550","Text":"and that was a 2x^3."},{"Start":"05:45.550 ","End":"05:47.715","Text":"Yes, sorry, it\u0027s cubed."},{"Start":"05:47.715 ","End":"05:52.305","Text":"We can divide everything by,"},{"Start":"05:52.305 ","End":"05:53.735","Text":"there I go again,"},{"Start":"05:53.735 ","End":"05:55.865","Text":"this has to be a 3 not a 5."},{"Start":"05:55.865 ","End":"05:57.710","Text":"Answer what we get."},{"Start":"05:57.710 ","End":"06:00.185","Text":"If we also reverse the order,"},{"Start":"06:00.185 ","End":"06:06.545","Text":"is this equation which really does fit the pattern that we were looking for,"},{"Start":"06:06.545 ","End":"06:09.440","Text":"where this is p,"},{"Start":"06:09.440 ","End":"06:11.555","Text":"this bit is q,"},{"Start":"06:11.555 ","End":"06:17.705","Text":"and this bit is r. Here\u0027s our differential equation."},{"Start":"06:17.705 ","End":"06:21.890","Text":"If you like, you could actually substitute each of x,"},{"Start":"06:21.890 ","End":"06:24.005","Text":"x^2, x^3 and see that it satisfies it."},{"Start":"06:24.005 ","End":"06:28.290","Text":"Anyway, here\u0027s the answer and we are done."}],"ID":7811}],"Thumbnail":null,"ID":4230}]

[{"ID":4228,"Videos":[7812,7813,7814,7815,7816,7817,7818,7819,7820,7821,7822,7823,7824,7825,7826,7827,7828,7829]},{"ID":4229,"Videos":[7804,7805,7806,7807,7808]},{"ID":4230,"Videos":[7809,7810,7811]}];

[7809,7810,7811];

1.1

1