Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:00,510 --> 00:00:04,190 Let's talk about solutions to differential equations. 2 00:00:04,470 --> 00:00:16,180 So solutions to these there's all types of solutions to ease of harm to all types of terminology. 3 00:00:16,230 --> 00:00:23,280 Say you have something like Y equals three X squared plus C. 4 00:00:23,280 --> 00:00:26,460 So this is called a one parameter 5 00:00:29,690 --> 00:00:43,760 one parameter family of solutions family of solutions so you have infinitely many solutions one for 6 00:00:43,760 --> 00:00:55,660 each choice of C say you had something like this C one each of the two x plus C to each of the five 7 00:00:55,660 --> 00:01:01,620 X. This is this would be called a two parameter family of solutions. 8 00:01:01,770 --> 00:01:15,160 So two parameter family solutions and you have infinitely many solutions again and all you have to do 9 00:01:15,160 --> 00:01:22,510 is you can pick the seasons infinitely many choices for the C's if you pick the C's say I pick C one 10 00:01:22,540 --> 00:01:28,990 equal to one that would give me either the two X and then C two equal to three that would give me each 11 00:01:28,990 --> 00:01:35,350 of the five X. This one doesn't have any constants and it doesn't have any arbitrary constants This 12 00:01:35,350 --> 00:01:39,190 is called a particular solution particular solution 13 00:01:42,020 --> 00:01:47,960 and it's free from arbitrary parameters that's why it's called the particular solution it has has no 14 00:01:47,960 --> 00:01:52,820 CS so free from arbitrary 15 00:01:55,370 --> 00:01:59,350 parameters. 16 00:02:00,360 --> 00:02:06,250 So if it has one C it's called a one parameter of family of solutions if it has two seas it's called 17 00:02:06,250 --> 00:02:12,430 a two parameter family of solutions if it doesn't have any CS It's called a particular solution and 18 00:02:12,450 --> 00:02:17,460 there's one more that's really important y equals zero. 19 00:02:17,460 --> 00:02:23,520 This is a special case whenever it's zero we call it the trivial solution so the trivial solution 20 00:02:26,040 --> 00:02:38,890 all of these are called explicit solutions explicit solutions because we have Y explicitly defined in 21 00:02:38,890 --> 00:02:40,580 terms of X. So. 22 00:02:40,640 --> 00:02:40,930 Right. 23 00:02:40,930 --> 00:02:48,760 They all look like Y equal stuff with X is only right. 24 00:02:48,760 --> 00:02:51,280 So we have specifically solved for wives. 25 00:02:51,280 --> 00:02:54,000 These are all explicit solutions. 26 00:02:54,130 --> 00:03:01,630 If Y is not explicitly given by a formula in terms of X or the other independent variable we'll just 27 00:03:01,630 --> 00:03:03,590 use X for clarity. 28 00:03:03,640 --> 00:03:06,800 You have what's called an implicit solution for example. 29 00:03:07,060 --> 00:03:17,760 We had Alan X plus Y minus Y squared equals each of the X plus X plus C. 30 00:03:17,830 --> 00:03:21,600 This here is not an explicit solution. 31 00:03:21,610 --> 00:03:25,230 This is called an implicit solution. 32 00:03:25,230 --> 00:03:28,500 So this is an implicit solution. 33 00:03:29,500 --> 00:03:33,030 OK so explicit means you you solve for y. 34 00:03:33,300 --> 00:03:37,530 Let's do a simple example to illustrate another key concept 35 00:03:41,990 --> 00:03:53,080 say we have this differential equation x y prime plus y equals zero and we want to show that Y equals 36 00:03:53,080 --> 00:03:56,200 one over X is a solution 37 00:03:59,930 --> 00:04:13,350 to this dy and we're also gonna find what's called the interval of definitions find the interval of 38 00:04:13,350 --> 00:04:14,130 definition 39 00:04:17,770 --> 00:04:18,240 okay. 40 00:04:18,280 --> 00:04:26,620 So to show it's a solution all we have to do is plug this function into this differential equation. 41 00:04:27,400 --> 00:04:30,610 So solution step are writing down the function again. 42 00:04:30,610 --> 00:04:33,200 So why is equal to 1 over x. 43 00:04:33,790 --> 00:04:36,760 And now we'll take the derivative of this to do that. 44 00:04:36,760 --> 00:04:43,270 You can write it as X to the negative one right you can take this and bring it upstairs and take the 45 00:04:43,270 --> 00:04:47,240 derivative you would use the power rule so you put the negative one in the front. 46 00:04:47,520 --> 00:04:51,220 If you get negative x the negative 1 minus 1 is negative 2. 47 00:04:51,270 --> 00:04:58,970 So you get negative 1 over X squared so y prime is equal to negative 1 over X squared. 48 00:04:59,720 --> 00:05:03,980 So now all we have to do is plug it into our differential equation. 49 00:05:03,980 --> 00:05:14,770 So we have X Y prime plus y sets equal to x and y prime we said was this right here. 50 00:05:14,900 --> 00:05:20,720 So this is negative 1 over X squared plus Y and then Y is right here. 51 00:05:21,590 --> 00:05:30,900 So 1 over X OK so all we've done is plug in the function and its derivative into our D axis cancels. 52 00:05:30,910 --> 00:05:35,620 We get negative one over x plus one over X and it's equal to zero. 53 00:05:35,620 --> 00:05:36,240 Hurrah. 54 00:05:36,250 --> 00:05:36,870 So it checks. 55 00:05:36,880 --> 00:05:41,410 We started with this and we showed it's equal to zero which is the deal here. 56 00:05:41,830 --> 00:05:43,300 So it is indeed a solution. 57 00:05:43,300 --> 00:05:45,890 That's what it means for a function to be a solution. 58 00:05:46,100 --> 00:05:46,310 OK. 59 00:05:46,330 --> 00:05:48,780 Now we have to find the interval of definition. 60 00:05:48,790 --> 00:05:55,180 So to do that I'm going to graph one over X so one over X looks like this. 61 00:05:55,180 --> 00:06:03,340 And it has a vertical asymptote at zero and a horizontal asymptote at zero. 62 00:06:03,340 --> 00:06:06,410 And it looks like this. 63 00:06:06,520 --> 00:06:09,410 And so we have to find what's called the interval of definition. 64 00:06:09,440 --> 00:06:16,840 Now the domain of this function is negative infinity to zero union zero to infinity. 65 00:06:16,840 --> 00:06:17,090 All right. 66 00:06:17,090 --> 00:06:19,440 So it's everything except zero. 67 00:06:19,670 --> 00:06:26,840 But that is not the interval of definition the interval of definition is the largest interval over which 68 00:06:26,840 --> 00:06:28,670 the solution is defined. 69 00:06:28,670 --> 00:06:31,750 So this here is not an interval OK. 70 00:06:31,940 --> 00:06:39,260 By definition interval is a set where if you pick any two numbers in that set every number between them 71 00:06:39,290 --> 00:06:43,160 is also in that set this set here does not satisfy that. 72 00:06:43,160 --> 00:06:43,420 Right. 73 00:06:43,430 --> 00:06:48,170 Because I can pick negative 1 and 2 and then zero fails. 74 00:06:48,170 --> 00:06:50,100 Right zero is not in the set. 75 00:06:50,120 --> 00:06:51,740 So this is not an interval. 76 00:06:51,770 --> 00:06:53,360 So what is the interval of definition. 77 00:06:53,360 --> 00:06:54,430 Well it's up to us. 78 00:06:54,440 --> 00:07:05,050 Actually we can pick this one so we can pick negative infinity to zero or we can pick this one so we 79 00:07:05,050 --> 00:07:06,820 can pick zero to infinity. 80 00:07:06,820 --> 00:07:12,390 So I'm just going to pick zero to infinity and we'll call that the interval of definition. 81 00:07:12,520 --> 00:07:16,780 So you get to pick whichever one you like most of the time. 82 00:07:18,010 --> 00:07:20,200 So I hope that made sense. 83 00:07:20,380 --> 00:07:25,740 Let's let's do one more example let's do one more example since we're talking about solutions. 84 00:07:25,960 --> 00:07:29,010 Let's finish talking about solutions there's one other key idea. 85 00:07:29,080 --> 00:07:35,750 Say we have this this differential equation d y the X equals X square root of Y. 86 00:07:36,290 --> 00:07:39,300 And what we know that a one parameter family of solutions 87 00:07:44,290 --> 00:07:46,900 family of solutions 88 00:07:49,400 --> 00:07:58,250 is given by this Y equals 1 fourth X squared plus C quantity squared. 89 00:07:58,700 --> 00:08:02,450 So this is a nonlinear D because we have that square root of Y. 90 00:08:02,660 --> 00:08:05,740 So why did the 1 1/2 power it's actually really easy to solve. 91 00:08:06,020 --> 00:08:15,340 And they're giving us the answer and the question is is there another 92 00:08:17,730 --> 00:08:18,390 solution 93 00:08:21,120 --> 00:08:24,480 so is there another solution to this differential equation. 94 00:08:24,480 --> 00:08:24,710 Right. 95 00:08:24,720 --> 00:08:28,430 That's not given by this one parameter family. 96 00:08:28,470 --> 00:08:29,590 So how would you figure that out. 97 00:08:29,620 --> 00:08:39,130 Well just observation the answer is yes and the answer is y equals zero Y equals zero is a solution 98 00:08:39,700 --> 00:08:41,250 to this differential equation. 99 00:08:41,260 --> 00:08:48,160 You can check if Y equals zero then d y the X while the derivative is also zero because it's a constant 100 00:08:48,160 --> 00:08:50,320 write the derivative of zero zero. 101 00:08:50,410 --> 00:08:58,000 So if you take Y equals zero and take d y the x equals zero and plug them back in here you get zero 102 00:08:58,030 --> 00:09:01,810 equals x times the square root of zero. 103 00:09:01,810 --> 00:09:03,420 So you get zero equals zero. 104 00:09:03,430 --> 00:09:04,210 Yep. 105 00:09:04,420 --> 00:09:06,040 So zero is a solution. 106 00:09:06,040 --> 00:09:11,260 So this is the solution that we can't get by picking C.. 107 00:09:11,270 --> 00:09:11,670 All right. 108 00:09:11,670 --> 00:09:13,380 We can't get Y equals. 109 00:09:13,390 --> 00:09:17,110 There's no there's no c we can pick that's going to make this zero. 110 00:09:17,200 --> 00:09:19,910 So this is called a singular solution. 111 00:09:20,080 --> 00:09:25,950 So whenever you have a solution that you can't get by picking values of C it's called a singular solution. 112 00:09:25,960 --> 00:09:27,700 So we can't get it. 113 00:09:27,730 --> 00:09:30,180 We cannot get this 114 00:09:32,970 --> 00:09:37,110 by picking C. It's impossible. 115 00:09:37,110 --> 00:09:43,320 So whenever you have a solution that you cannot get by picking values of C it is called a singular solution 116 00:09:43,320 --> 00:09:44,740 super super important. 117 00:09:45,390 --> 00:09:50,880 If it turns out to be the case that you can get all the solutions by picking values of C you have what's 118 00:09:50,880 --> 00:09:53,090 called the general solution let me write that down. 119 00:09:53,090 --> 00:09:54,780 It's really important. 120 00:09:54,780 --> 00:09:57,870 So if we can get all solutions 121 00:10:00,730 --> 00:10:09,750 that you can get every single solution by picking C we have what's called the general solution. 122 00:10:09,750 --> 00:10:15,630 We have the general solution. 123 00:10:15,870 --> 00:10:21,660 So in this case this one parameter family here is not the general solution. 124 00:10:21,660 --> 00:10:21,900 Right. 125 00:10:21,900 --> 00:10:26,740 Because here is a solution that we could not get by picking values of seat. 126 00:10:27,030 --> 00:10:28,110 Hence the name singular. 127 00:10:28,110 --> 00:10:29,740 So it's a singular solution. 128 00:10:29,740 --> 00:10:30,560 All right. 129 00:10:30,570 --> 00:10:34,410 If it were the case that we can get all the solutions then we would have what was called the general 130 00:10:34,410 --> 00:10:35,460 solution. 131 00:10:35,460 --> 00:10:41,430 Whenever you have a linear differential equation you always have the general solution that's a big big 132 00:10:41,430 --> 00:10:44,480 result from differential equations. 133 00:10:44,520 --> 00:10:49,830 I hope that video made sense with a lot of information and in the next video we'll briefly talk about 134 00:10:49,830 --> 00:10:51,390 what's called initial Valley problems. 135 00:10:51,390 --> 00:10:51,720 That's it. 13176