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These are the user uploaded subtitles that are being translated: 1 00:00:00,780 --> 00:00:02,080 Hey everyone welcome back. 2 00:00:02,090 --> 00:00:08,170 And this video we're going to talk about the difference between what's called linear differential equations 3 00:00:08,680 --> 00:00:13,990 vs. nonlinear linear versus non-linear. 4 00:00:14,020 --> 00:00:17,040 This topic is a little bit harder for some people. 5 00:00:17,050 --> 00:00:19,680 So we'll do a lot of examples. 6 00:00:19,780 --> 00:00:28,480 So A D E so a differential equation is linear in Y. 7 00:00:28,500 --> 00:00:35,800 You can also say linear in X or any other variable we're going to focus on linear in y if we can write 8 00:00:35,800 --> 00:00:36,040 it. 9 00:00:37,000 --> 00:00:44,290 So if we can write it in the form. 10 00:00:44,450 --> 00:00:48,080 So first I'm going to show you you like the mathematical definition and then I'm going to explain it 11 00:00:48,080 --> 00:00:50,330 in words because that's how I think of it. 12 00:00:50,330 --> 00:00:52,340 I actually don't think of it this way. 13 00:00:52,340 --> 00:00:53,650 I mean I guess I do. 14 00:00:54,320 --> 00:00:55,760 Let me just go ahead and write it. 15 00:00:55,880 --> 00:00:58,010 So basically you have a coefficient. 16 00:00:58,160 --> 00:01:10,610 So I'll call that a sub end of X so a pure function of x times say the derivative of x plus and then 17 00:01:10,640 --> 00:01:13,040 you would have a sub N minus one. 18 00:01:13,040 --> 00:01:16,940 But I'll skip that one all the way down to say the first derivative. 19 00:01:17,090 --> 00:01:28,060 So a sub 1 of X D Y D X again a pure function of x and then plus a sub zero of X times the zero with 20 00:01:28,060 --> 00:01:28,960 derivative. 21 00:01:29,140 --> 00:01:37,180 So the zero is derivative of a function is the original function and that should be equal to G of X. 22 00:01:37,190 --> 00:01:42,190 So basically in front of y and all of its derivatives. 23 00:01:42,200 --> 00:01:43,310 So here's why. 24 00:01:43,940 --> 00:01:48,520 And all of its derivatives you only have pure functions of x. 25 00:01:48,860 --> 00:01:50,690 That's the first condition. 26 00:01:50,720 --> 00:01:56,570 So again in front of Y in all of its derivatives you only have pure functions of x and the second condition 27 00:01:56,570 --> 00:02:03,740 is that y in all of its derivatives are all to the first power so maybe you should write that down so 28 00:02:03,740 --> 00:02:05,360 1 so in front 29 00:02:08,170 --> 00:02:11,710 of y and its derivatives 30 00:02:16,890 --> 00:02:22,790 you have only pure functions of acts so pure functions of x. 31 00:02:22,840 --> 00:02:24,710 Also here you have the only X here. 32 00:02:24,720 --> 00:02:28,920 Right can't have a Y here but in front of y and all of its derivatives you have pure functions of x 33 00:02:29,010 --> 00:02:31,980 and then also y and its derivatives 34 00:02:35,620 --> 00:02:37,330 are altered the first power. 35 00:02:37,330 --> 00:02:48,700 So are all to the first power OK let's do some examples right away and we're gonna do a bunch of these 36 00:02:48,810 --> 00:02:51,900 and so hopefully it makes sense. 37 00:02:51,950 --> 00:02:52,510 It's a question. 38 00:02:52,510 --> 00:02:57,180 It says linear or nonlinear so linear in Y. 39 00:02:57,190 --> 00:03:03,550 We're only concerned if it's linear or non-linear and Y or is it nonlinear nonlinear. 40 00:03:03,630 --> 00:03:04,780 OK. 41 00:03:04,860 --> 00:03:08,240 A How about let's say I have to make these up. 42 00:03:08,240 --> 00:03:11,070 How about y double prime plus. 43 00:03:11,070 --> 00:03:17,290 Each of the X times Y prime plus Y equals sine X. 44 00:03:17,400 --> 00:03:19,030 So this is a differential equation. 45 00:03:19,080 --> 00:03:24,480 If you watch the first video you'll know that the order is two because the order is the order of the 46 00:03:24,480 --> 00:03:27,660 highest derivative and there's a second derivative in this problem. 47 00:03:27,660 --> 00:03:30,040 We want to figure out if it's linear or nonlinear. 48 00:03:30,060 --> 00:03:31,700 So here we have a one. 49 00:03:32,160 --> 00:03:33,370 Here we have an e of the X. 50 00:03:33,390 --> 00:03:36,900 Here we have a one so in front of y and all of its derivatives. 51 00:03:36,900 --> 00:03:42,110 We have pure functions of x that that's the first condition check and Y. 52 00:03:42,870 --> 00:03:45,590 And all of its derivatives are all to the first power. 53 00:03:45,930 --> 00:03:47,630 So that's the second condition check. 54 00:03:47,640 --> 00:03:50,640 And then here we have a pure function of x so check. 55 00:03:50,640 --> 00:03:54,000 So in this case the answer would be linear linear. 56 00:03:54,060 --> 00:03:57,060 Let's do another one. 57 00:03:57,060 --> 00:04:10,550 B How about this one y triple prime plus three y y double prime plus Y equals each of the x. 58 00:04:10,960 --> 00:04:12,160 In this case the order is three. 59 00:04:12,160 --> 00:04:16,130 Just easy to talk about because the order is the order of the highest derivative. 60 00:04:16,360 --> 00:04:22,060 And this one would be nonlinear because of this piece right here. 61 00:04:22,060 --> 00:04:22,300 Right. 62 00:04:22,300 --> 00:04:26,140 This is a function of y and it's being multiplied by the second derivative. 63 00:04:26,140 --> 00:04:27,920 You cannot have that right. 64 00:04:27,940 --> 00:04:32,020 You have to have only pure functions of x in front of y and its derivatives. 65 00:04:32,020 --> 00:04:37,250 So in this case the answer would be nonlinear nonlinear. 66 00:04:37,250 --> 00:04:39,300 Let's do another one. 67 00:04:39,310 --> 00:04:42,840 C How about something like this. 68 00:04:42,870 --> 00:04:50,060 X squared away the X plus Y equals seven OK. 69 00:04:50,120 --> 00:04:54,650 So in this case the order is one because the order is the order of the highest derivative. 70 00:04:54,650 --> 00:04:59,720 So we have the first derivative and this one won't be linear because this is a pure function of X. There 71 00:04:59,720 --> 00:05:00,460 is a one here. 72 00:05:00,470 --> 00:05:05,670 It's a pure function of x and y and all of its derivatives are to the first power. 73 00:05:05,740 --> 00:05:08,350 So that would be linear. 74 00:05:08,780 --> 00:05:13,550 Keep going D How about something like this square root. 75 00:05:13,700 --> 00:05:23,950 Do you y the x plus three y equals sine X in this case the order is 1 because this is the first derivative 76 00:05:24,370 --> 00:05:25,660 and this one is nonlinear. 77 00:05:25,660 --> 00:05:32,680 And the reason is you can think of this D Whitey X this piece here rather as do y the X to the one half 78 00:05:32,680 --> 00:05:33,500 power right. 79 00:05:33,520 --> 00:05:35,760 Because you can write the square root as a 1 1/2. 80 00:05:36,430 --> 00:05:38,920 So that's not the first power. 81 00:05:38,920 --> 00:05:41,070 So this is non-linear right. 82 00:05:41,260 --> 00:05:47,120 Y and all of its derivatives all have to be to these first powers. 83 00:05:47,200 --> 00:05:58,450 Otherwise it's nonlinear e y double prime plus sine y times Y equals E to the. 84 00:05:58,510 --> 00:06:04,210 This wouldn't be order to right because the order is the order of the highest derivative and it would 85 00:06:04,210 --> 00:06:07,770 be nonlinear because you have this function of y here in front of the Y. 86 00:06:07,790 --> 00:06:08,820 He can't have that right. 87 00:06:08,830 --> 00:06:10,280 No good at all. 88 00:06:10,330 --> 00:06:18,420 So this is non-linear you can only have pure functions of x in front of y and all of its derivatives. 89 00:06:18,430 --> 00:06:24,310 One more f see if I have any ideas I can't really think of anything that we haven't really done too 90 00:06:24,310 --> 00:06:25,440 much. 91 00:06:25,460 --> 00:06:33,780 Well how about this Y double prime plus cosine of X plus Y equals each of the x. 92 00:06:33,860 --> 00:06:40,320 So kind of a weird problem here but this is a differential equation and it's order 2 because the order 93 00:06:40,320 --> 00:06:44,560 is the order of the highest derivative and that's a second derivative it's been non-linear. 94 00:06:44,570 --> 00:06:50,410 Right non-linear you cannot have cosine of X plus Y doesn't make any sense in this context right. 95 00:06:50,410 --> 00:06:56,450 Y and all of its derivatives have to be to the first power it's inside a cosine function so it's not 96 00:06:56,450 --> 00:06:57,470 good. 97 00:06:57,590 --> 00:06:59,220 That would be non. 98 00:06:59,300 --> 00:07:03,770 So hopefully this medium video made sense and just keep in mind it's linear. 99 00:07:03,770 --> 00:07:09,500 If Y and all of its derivatives appear to the first power and in front of y and all of its derivatives 100 00:07:09,890 --> 00:07:16,790 you have to have pure functions of x on the right hand side you always have a pure function of X.. 101 00:07:16,820 --> 00:07:17,390 I hope that helps. 9850