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These are the user uploaded subtitles that are being translated: 1 00:00:00,300 --> 00:00:01,480 Hey everyone. 2 00:00:01,500 --> 00:00:10,380 And this video we have a differential equation y prime plus two X Y squared equals zero. 3 00:00:10,410 --> 00:00:15,490 So this is a non linear differential equation because of the Y squared. 4 00:00:15,780 --> 00:00:18,060 And we're told that a one parameter 5 00:00:21,300 --> 00:00:26,950 family of solutions so family of solutions 6 00:00:30,760 --> 00:00:40,240 is given by y equals one over X squared plus C and the questions are to find 7 00:00:42,930 --> 00:00:52,710 the solutions or find the solution rather that passes through the given point. 8 00:00:52,730 --> 00:01:10,320 So through the given point and find the interval of definition the interval of definition 9 00:01:13,580 --> 00:01:15,920 part a let's go through this very carefully. 10 00:01:15,920 --> 00:01:18,270 This is a really key example. 11 00:01:18,470 --> 00:01:25,030 So maybe the solution passes through zero negative 1 So again the question is to find a solution that 12 00:01:25,030 --> 00:01:29,080 passes through this point and find the interval of definition. 13 00:01:29,620 --> 00:01:34,030 So solution so in this case x is 0 and Y is negative 1. 14 00:01:34,040 --> 00:01:37,070 So x is 0 and Y is negative 1. 15 00:01:37,100 --> 00:01:39,410 So all we have to do is we have to find c. 16 00:01:39,590 --> 00:01:49,470 So we're gonna put the negative one where the y is right here and that's equal to 1 over 0 squared plus 17 00:01:49,470 --> 00:01:52,770 C which is putting the zero here where the X is. 18 00:01:52,780 --> 00:01:59,730 So that means that we have negative 1 equals 1 oversee to solve this perceived just multiply by C so 19 00:01:59,780 --> 00:02:01,730 you get negative C equals 1. 20 00:02:01,860 --> 00:02:04,740 So c is equal to negative 1. 21 00:02:04,850 --> 00:02:05,900 We're almost there. 22 00:02:05,900 --> 00:02:14,480 All we have to do is plug the C back into our 1 parameter family so we get Y equals 1 over x squared 23 00:02:15,080 --> 00:02:16,220 minus 1. 24 00:02:16,250 --> 00:02:21,570 So that is the solution to the D E that passes through this point. 25 00:02:21,680 --> 00:02:24,130 You can think of this as an initial condition. 26 00:02:24,170 --> 00:02:27,930 This is why of zero equals negative one. 27 00:02:28,720 --> 00:02:31,670 OK so now we just have to find the interval of definition. 28 00:02:31,700 --> 00:02:40,250 So to do that what you can do is you can graph this craft this solution wrap this function rather. 29 00:02:40,430 --> 00:02:41,780 So how do you graft this function. 30 00:02:41,780 --> 00:02:42,850 Well skill. 31 00:02:43,450 --> 00:02:45,670 So this function has two vertical asymptote. 32 00:02:45,670 --> 00:02:55,500 I'll be brief one in negative one so there's 1 and then here's negative 1. 33 00:02:55,670 --> 00:03:01,940 This is one and this is negative one and it has a horizontal asymptote at zero. 34 00:03:01,970 --> 00:03:06,940 That's because the degree in the denominator is bigger than the degree and the numerator. 35 00:03:06,940 --> 00:03:09,870 So it has a horizontal asymptote but zero. 36 00:03:10,100 --> 00:03:12,890 And if you plug in zero you get negative one. 37 00:03:12,890 --> 00:03:16,980 In other words it passes through zero come a negative 1. 38 00:03:17,060 --> 00:03:22,200 So it looks something like this and it looks something like this. 39 00:03:22,210 --> 00:03:27,220 You can use your calculator to it's pretty easy to plug in numbers and figure out like you plug in 5. 40 00:03:27,330 --> 00:03:29,850 If you plug in 5 you'll notice it's up here. 41 00:03:29,850 --> 00:03:31,690 So it has to look like that. 42 00:03:31,690 --> 00:03:35,490 If you plug a negative 10 you'll notice the y values up here so you know it has to look like that. 43 00:03:36,140 --> 00:03:39,300 OK so what is the interval of definition. 44 00:03:39,750 --> 00:03:43,180 Well we talked about this before in a previous video. 45 00:03:43,370 --> 00:03:49,670 The domain here would be negative infinity to negative 1 negative 1 to 1 1 to infinity with union symbols 46 00:03:49,680 --> 00:03:50,570 between them. 47 00:03:50,950 --> 00:03:54,470 We would exclude negative 1 to 1 but an interval can't have that right. 48 00:03:54,480 --> 00:03:57,330 Interval cannot have gaps. 49 00:03:57,330 --> 00:04:03,070 So the answer is going to be in this case negative 1 to 1 right here. 50 00:04:03,330 --> 00:04:10,140 And the reason that that's the answer is because that is where the point is right. 51 00:04:10,140 --> 00:04:13,040 The graph passes through zero negative 1. 52 00:04:13,110 --> 00:04:14,720 So we want to pick this. 53 00:04:14,730 --> 00:04:18,180 So this here is blue graph that I'm drawing. 54 00:04:18,180 --> 00:04:19,840 That's the graph of the solution. 55 00:04:19,920 --> 00:04:20,580 Right. 56 00:04:20,610 --> 00:04:21,870 And so where is that defined. 57 00:04:21,870 --> 00:04:24,000 On negative 1 to 1. 58 00:04:24,000 --> 00:04:25,390 Let's do another example. 59 00:04:25,410 --> 00:04:29,760 So if it had been like over here or something if the point had been here then we would have picked one 60 00:04:29,760 --> 00:04:30,870 to infinity. 61 00:04:30,870 --> 00:04:31,640 Check this one out. 62 00:04:31,920 --> 00:04:38,680 Let's say it passes through to one third so a solution. 63 00:04:38,680 --> 00:04:42,870 The same thing your your X is 2 and your Y is one third. 64 00:04:43,460 --> 00:04:55,210 So you get one third equals one over two squared plus C plugging it into our R function and so we get 65 00:04:55,210 --> 00:05:03,100 one third equals one over four plus C then we can cross multiply so we get. 66 00:05:03,130 --> 00:05:05,310 So this goes here and this goes here. 67 00:05:05,530 --> 00:05:10,150 So we get 4 plus C equals 3. 68 00:05:10,360 --> 00:05:11,520 Subtract 4. 69 00:05:11,890 --> 00:05:14,060 So we get C equals negative 1 Hurrah. 70 00:05:14,070 --> 00:05:15,760 So we get exactly the same thing. 71 00:05:15,760 --> 00:05:17,620 It's totally rigged. 72 00:05:17,620 --> 00:05:19,240 So we already have the graph. 73 00:05:19,780 --> 00:05:23,320 And in this case I'll read I'll redraw that well I'll just use this graph. 74 00:05:23,320 --> 00:05:24,150 Here's 2 1. 75 00:05:24,280 --> 00:05:27,990 To one third right. 76 00:05:28,020 --> 00:05:31,690 There's two one third so this time we use this piece here and the purple piece. 77 00:05:31,710 --> 00:05:35,010 So now the interval of definition would be 1 to infinity. 78 00:05:35,010 --> 00:05:40,590 So you have to pick the interval from where the point is in the first example it was here. 79 00:05:41,010 --> 00:05:44,350 So we pick negative 1 to 1 and the second example it was here. 80 00:05:44,520 --> 00:05:45,680 So we picked one to infinity. 81 00:05:45,690 --> 00:05:50,630 If it had been over here on the left we would pick negative infinity to negative 1. 82 00:05:50,670 --> 00:05:53,870 Let's do one more one more really really tricky one. 83 00:05:55,270 --> 00:06:03,490 What if we're told it passes through five comma zero then again that's just like before we do the same 84 00:06:03,490 --> 00:06:03,760 thing. 85 00:06:03,760 --> 00:06:05,700 This is our X. This is our y. 86 00:06:05,950 --> 00:06:13,560 So we get 0 equals 1 over 5 squared plus C.. 87 00:06:13,850 --> 00:06:20,900 So zero is equal to one over twenty five plus C then multiply by twenty five policy. 88 00:06:21,740 --> 00:06:23,780 So you get right. 89 00:06:23,810 --> 00:06:28,400 You would you would put that here like this and do that here like this. 90 00:06:28,430 --> 00:06:31,510 And then what happens is these cancel so you get zero equals 1. 91 00:06:31,520 --> 00:06:32,950 So there is no solution. 92 00:06:33,000 --> 00:06:34,070 All right there is no c. 93 00:06:34,760 --> 00:06:36,050 So does that mean there is no answer. 94 00:06:36,050 --> 00:06:36,480 No. 95 00:06:36,520 --> 00:06:37,130 No it does not. 96 00:06:37,130 --> 00:06:42,290 The question is to find the solution that passes through the point someone to rewrite the D here. 97 00:06:42,650 --> 00:06:46,280 The D was my prime plus 2 x y squared. 98 00:06:46,280 --> 00:06:54,460 That was our original question and our one parameter family was won over X squared plus C we tried to 99 00:06:54,460 --> 00:06:58,780 find the solution that passed this point and we got something ridiculous. 100 00:06:59,200 --> 00:07:00,340 So what is the solution. 101 00:07:00,370 --> 00:07:05,490 Well now we have to think Is there another solution to this differential equation. 102 00:07:05,550 --> 00:07:12,750 Right there is y equals zero because if you let Y equal to zero then y prime is also zero. 103 00:07:13,920 --> 00:07:18,930 And you can check that this indeed does satisfy the differential equation plugging it in. 104 00:07:18,930 --> 00:07:25,000 You get 0 plus 2 x 0 squared so zero zero zero. 105 00:07:25,010 --> 00:07:26,070 So it checks. 106 00:07:26,070 --> 00:07:27,990 So y equals zero is a solution. 107 00:07:27,990 --> 00:07:31,080 So the answer is y equals zero. 108 00:07:31,110 --> 00:07:32,230 That is a solution. 109 00:07:32,370 --> 00:07:35,490 And it does indeed pass through five comma zero Y. 110 00:07:35,520 --> 00:07:45,290 Well if you graph it y equals zero as a horizontal line and there is five comma zero so it certainly 111 00:07:45,290 --> 00:07:47,480 passes through five comma zero. 112 00:07:47,480 --> 00:07:51,090 So y equals zero is the solution to this differential equation. 113 00:07:51,290 --> 00:07:52,990 And it does pass through five comma zero. 114 00:07:53,000 --> 00:07:56,980 So as it is the answer to this question what about the interval of definition. 115 00:07:56,990 --> 00:08:02,770 Well this thing is defined everywhere so it would just be negative infinity to infinity. 116 00:08:02,850 --> 00:08:03,580 Hope that made sense. 10978