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These are the user uploaded subtitles that are being translated: 1 00:00:00,530 --> 00:00:06,960 Everyone in this video were briefly going to talk about what's called initial value problems so initial 2 00:00:07,710 --> 00:00:11,750 value problems. 3 00:00:11,800 --> 00:00:16,000 I just wanna explain the concept behind them and what they mean. 4 00:00:16,160 --> 00:00:17,150 So high. 5 00:00:17,160 --> 00:00:21,380 V.P. is an easy way to abbreviate initial value problem. 6 00:00:21,430 --> 00:00:22,410 Due to examples. 7 00:00:22,430 --> 00:00:29,510 So Example 1 say we have a differential equation so d y the X and it's equal to some stuff over here 8 00:00:29,510 --> 00:00:37,030 on the right hand side with Xs and Ys so to indicate that I'm gonna to write F of X comma Y. 9 00:00:37,040 --> 00:00:46,940 So all this means is that we have some stuff with X's and Y's that's all that notation means. 10 00:00:46,950 --> 00:00:47,180 Okay. 11 00:00:47,220 --> 00:00:55,980 So like X plus Y etc. X squared plus sine Y etc. I just stop with X and Y then we have a condition Y 12 00:00:56,010 --> 00:01:00,090 of x sub zero equals Y some zero. 13 00:01:00,090 --> 00:01:02,020 This is called an initial condition. 14 00:01:02,040 --> 00:01:10,080 So initial condition and typically I like to revisit it as I see. 15 00:01:10,290 --> 00:01:13,270 So that's our I see s our initial condition. 16 00:01:13,310 --> 00:01:14,570 So what does this mean graphically. 17 00:01:14,600 --> 00:01:19,790 Well if you solve a differential equation you know most of the time you get like you know X squared 18 00:01:19,850 --> 00:01:23,050 plus C or something like that you get a constant. 19 00:01:23,090 --> 00:01:27,890 In this case you would get one arbitrary constant so you might get like you know infinitely many solutions 20 00:01:27,900 --> 00:01:29,030 so something like this. 21 00:01:29,030 --> 00:01:33,920 And then something like this something like this one for each choice of C right you have infinitely 22 00:01:33,920 --> 00:01:34,720 many solutions. 23 00:01:35,570 --> 00:01:41,480 And so what's happening here is that this initial condition well what does this mean. 24 00:01:41,480 --> 00:01:46,290 This means that the x coordinate is X not on the y coordinate is why not. 25 00:01:46,290 --> 00:01:49,430 So this means that your solution passes this point. 26 00:01:49,430 --> 00:01:57,810 So what it's doing is it's picking a particulates solution from the infinitely many solutions. 27 00:01:57,810 --> 00:02:00,370 So like maybe it's picking this one. 28 00:02:00,480 --> 00:02:03,990 So this red graph here this this would be the solution 29 00:02:07,180 --> 00:02:08,110 to the AVP 30 00:02:10,920 --> 00:02:17,070 so whenever you have one arbitrary constant and your solution you haven't wanted whenever you have a 31 00:02:17,070 --> 00:02:22,380 one parameter family to find a particular solution you would need one initial condition. 32 00:02:22,590 --> 00:02:28,170 In this case the initial condition tells us graphically that our solution passes the point. 33 00:02:28,170 --> 00:02:32,010 So we have infinitely many solutions and then we're picking a particular one. 34 00:02:32,010 --> 00:02:32,820 Right. 35 00:02:32,880 --> 00:02:34,920 Maybe that's what the name particular comes from right. 36 00:02:34,920 --> 00:02:40,390 Because once you use this you're gonna find your C so you'll get something like Y equals. 37 00:02:40,530 --> 00:02:43,680 Making this up but X squared plus four let's say. 38 00:02:43,680 --> 00:02:45,070 So that might be your answer. 39 00:02:45,090 --> 00:02:48,620 So this is called a particular solution because it has no C's. 40 00:02:48,630 --> 00:02:50,640 So that's that's an initial value problem. 41 00:02:50,670 --> 00:02:52,040 Here's another example. 42 00:02:52,140 --> 00:02:59,480 Say we had DIY D X equal to now actually mean do a second derivative here. 43 00:02:59,520 --> 00:03:00,280 There we go. 44 00:03:00,540 --> 00:03:04,880 Equal to some stuff with X Y and the first derivative. 45 00:03:04,890 --> 00:03:10,740 So now we have an order to differential equation so X Y and then the first derivative. 46 00:03:10,740 --> 00:03:15,420 So this is just some stuff with X Y and the first derivative. 47 00:03:16,680 --> 00:03:26,670 So now we need to initial conditions maybe y of x sub zero equals Y sub zero and then Y prime of x sub 48 00:03:26,680 --> 00:03:31,870 zero equals Y sub one equals Y sub 1. 49 00:03:31,870 --> 00:03:35,830 So now again say we have you know infinitely many solutions you know something like this 50 00:03:39,270 --> 00:03:39,520 right. 51 00:03:39,520 --> 00:03:46,450 And then what this does is it obviously it picks the one that passes through through X not why not. 52 00:03:46,450 --> 00:03:48,210 So it picks that one. 53 00:03:48,220 --> 00:03:55,420 So maybe that would be this one here so it would pick that one but not only that it guarantees that 54 00:03:55,420 --> 00:04:06,610 the slope at this point is is this right this slope slope is why 1 the slope of this tangent line is 55 00:04:06,640 --> 00:04:11,290 why once the slope of the function or the tangent line at that point is equal to that. 56 00:04:11,290 --> 00:04:12,980 So this requires two things. 57 00:04:13,000 --> 00:04:13,270 Right. 58 00:04:13,270 --> 00:04:18,530 It passes through this point and it has this particular slope at that point. 59 00:04:18,560 --> 00:04:23,990 So in this case if like if you were to solve this d e maybe it would be something like I'm just gonna 60 00:04:24,010 --> 00:04:25,120 make this up. 61 00:04:25,120 --> 00:04:29,470 C 1 each of the X plus C 2 x squared was totally making this up. 62 00:04:29,630 --> 00:04:35,380 So you'd have to seize and see what you would have to use both of these conditions to find both CS and 63 00:04:35,380 --> 00:04:38,050 when you do that you would get the particular solution. 64 00:04:38,050 --> 00:04:43,060 So in this case the particular solution would be would be that right there. 65 00:04:43,180 --> 00:04:45,490 So I hope that made sense. 66 00:04:45,490 --> 00:04:51,250 I'm going to make another video showing you a little bit more how to how to actually find the seas and 67 00:04:51,250 --> 00:04:52,030 stuff like that. 68 00:04:52,030 --> 00:04:52,370 That's it. 6881