1 00:00:00,000 --> 00:00:01,010 2 00:00:01,010 --> 00:00:04,120 Let's do one more homogeneous differential equation, or 3 00:00:04,120 --> 00:00:06,189 first order homogeneous differential equation, to 4 00:00:06,190 --> 00:00:08,530 differentiate it from the homogeneous linear 5 00:00:08,529 --> 00:00:09,809 differential equations we'll do later. 6 00:00:09,810 --> 00:00:12,040 But anyway, the problem we have here. 7 00:00:12,039 --> 00:00:18,109 It's the derivative of y with respect to x is equal to-- 8 00:00:18,109 --> 00:00:23,550 that x looks like a y-- is equal to x 9 00:00:23,550 --> 00:00:28,089 squared plus 3y squared. 10 00:00:28,089 --> 00:00:30,312 I'm writing it a little bit small, so that I 11 00:00:30,312 --> 00:00:31,970 don't run out of space. 12 00:00:31,969 --> 00:00:35,670 Divided by 2xy. 13 00:00:35,670 --> 00:00:39,270 So in all of these homogeneous equations-- and obviously, we 14 00:00:39,270 --> 00:00:40,370 don't know it's homogeneous yet. 15 00:00:40,369 --> 00:00:43,439 So we have to try to write it as a function of 16 00:00:43,439 --> 00:00:44,909 y divided by x. 17 00:00:44,909 --> 00:00:46,929 So it looks like we could do that, if we divide the top and 18 00:00:46,929 --> 00:00:48,109 the bottom by x squared. 19 00:00:48,109 --> 00:00:50,450 So if we just multiply-- let me do it in a different 20 00:00:50,450 --> 00:00:54,370 color-- 1 over x squared, or x to the negative 2, 21 00:00:54,369 --> 00:00:56,239 over 1 over x squared. 22 00:00:56,240 --> 00:00:58,370 We're essentially just multiplying by 1. 23 00:00:58,369 --> 00:01:00,859 And then, that is equal to what? 24 00:01:00,859 --> 00:01:11,370 1 plus 3y squared over x squared divided by 2-- if you 25 00:01:11,370 --> 00:01:14,109 divide x, divide by x squared, you just get a 1 over x-- so 2 26 00:01:14,109 --> 00:01:16,609 times y over x. 27 00:01:16,609 --> 00:01:19,730 Or we could just rewrite the whole thing, and this is just 28 00:01:19,730 --> 00:01:27,930 equal to 1 plus 3y over x squared divided by 2 29 00:01:27,930 --> 00:01:29,260 times y over x. 30 00:01:29,260 --> 00:01:31,840 So yes, this is a homogeneous equation. 31 00:01:31,840 --> 00:01:34,260 Because we were able to write it as a function of 32 00:01:34,260 --> 00:01:35,490 y divided by x. 33 00:01:35,489 --> 00:01:37,569 So now, we can do the substitution with v. 34 00:01:37,569 --> 00:01:39,199 And hopefully, this is starting to become a little 35 00:01:39,200 --> 00:01:40,750 bit of second nature to you. 36 00:01:40,750 --> 00:01:43,609 So we can make the substitution that v is equal 37 00:01:43,609 --> 00:01:47,719 to y over x, or another way of writing that is that y is 38 00:01:47,719 --> 00:01:50,659 equal to xv. 39 00:01:50,659 --> 00:01:53,149 And then, of course, the derivative of y with respect 40 00:01:53,150 --> 00:01:56,130 to x, or if we take the derivative with respect to x 41 00:01:56,129 --> 00:01:59,099 of both sides of this, that's equal to the derivative of x 42 00:01:59,099 --> 00:02:05,059 is 1 times v, this is just the product rule, plus x times the 43 00:02:05,060 --> 00:02:09,050 derivative of v with respect to x. 44 00:02:09,050 --> 00:02:11,545 And now, we can substitute the derivative of y with respect 45 00:02:11,544 --> 00:02:13,489 to x is just this. 46 00:02:13,490 --> 00:02:15,270 And then the right hand side of the equation is this. 47 00:02:15,270 --> 00:02:17,360 But we can substitute v for y over x. 48 00:02:17,360 --> 00:02:19,230 So let's do that. 49 00:02:19,229 --> 00:02:24,179 And so we get v plus x. 50 00:02:24,180 --> 00:02:25,020 Instead of dv [? dv ?] 51 00:02:25,020 --> 00:02:27,570 x, I'll write v prime for now, just so that I don't take up 52 00:02:27,569 --> 00:02:29,150 too much space. 53 00:02:29,150 --> 00:02:36,420 v prime is equal to 1 plus 3v squared, we're making the 54 00:02:36,419 --> 00:02:38,474 substitution v is equal to y over x. 55 00:02:38,474 --> 00:02:41,799 56 00:02:41,800 --> 00:02:43,140 All of that over 2v. 57 00:02:43,139 --> 00:02:46,319 58 00:02:46,319 --> 00:02:47,289 Now, let's see what we can do. 59 00:02:47,289 --> 00:02:49,969 This is where we just get our algebra hat on, and try to 60 00:02:49,969 --> 00:02:53,990 simplify until it's a separable equation in v. 61 00:02:53,990 --> 00:02:54,860 So let's do that. 62 00:02:54,860 --> 00:02:57,950 So let's multiply both sides of this equation by 2v. 63 00:02:57,949 --> 00:03:08,250 So we'll get 2v squared plus 2xv v prime-- 2v times x, yep, 64 00:03:08,250 --> 00:03:14,569 that's 2xv v prime-- is equal to 1 plus 3v squared. 65 00:03:14,569 --> 00:03:16,759 Now let's see, let's subtract 2v squared from 66 00:03:16,759 --> 00:03:18,349 both sides of this. 67 00:03:18,349 --> 00:03:29,579 And we will be left with 2xv v prime is equal to 1 plus-- 68 00:03:29,580 --> 00:03:31,700 let's see, we're subtracting 2v squared from both sides. 69 00:03:31,699 --> 00:03:34,519 So we're just left with a 1 plus v squared here, right? 70 00:03:34,520 --> 00:03:37,800 3v squared minus 2v squared is just v squared. 71 00:03:37,800 --> 00:03:40,080 And let's see, we want it to be separable, so let's put all 72 00:03:40,080 --> 00:03:42,420 the v's on the left hand side. 73 00:03:42,419 --> 00:03:51,899 So we get 2xv v prime divided by 1 plus v 74 00:03:51,900 --> 00:03:55,640 squared is equal to 1. 75 00:03:55,639 --> 00:03:58,349 And let's divide both sides by x. 76 00:03:58,349 --> 00:04:00,099 So we get the x's on the other side. 77 00:04:00,099 --> 00:04:05,210 So then we get 2v-- and I'll now switch back 78 00:04:05,210 --> 00:04:06,099 to the other notation. 79 00:04:06,099 --> 00:04:11,519 Instead of v prime, I'll write dv dx. 80 00:04:11,520 --> 00:04:15,760 2v times the derivative of v with respect to x divided by 1 81 00:04:15,759 --> 00:04:20,139 plus v squared is equal to-- I'm dividing both sides by x, 82 00:04:20,139 --> 00:04:22,199 notice I didn't write the x on this side-- so that is equal 83 00:04:22,199 --> 00:04:24,560 to 1 over x. 84 00:04:24,560 --> 00:04:27,970 And then, if we just multiply both sides of this times dx, 85 00:04:27,970 --> 00:04:30,310 we've separated the two variables and we can integrate 86 00:04:30,310 --> 00:04:30,939 both sides. 87 00:04:30,939 --> 00:04:32,709 So let's do that. 88 00:04:32,709 --> 00:04:33,529 Let's go up here. 89 00:04:33,529 --> 00:04:35,289 I'll switch to a different color, so you know I'm working 90 00:04:35,290 --> 00:04:37,540 on a different column now. 91 00:04:37,540 --> 00:04:39,400 So multiply both sides by dx. 92 00:04:39,399 --> 00:04:51,049 I get 2v over 1 plus v squared dv is equal to 1 over x dx. 93 00:04:51,050 --> 00:04:54,639 And now we can just integrate both sides of this equation. 94 00:04:54,639 --> 00:04:57,959 This is a separable equation in terms of v and x. 95 00:04:57,959 --> 00:04:58,979 And what's the integral of this? 96 00:04:58,980 --> 00:05:00,860 At first, you might think, oh boy, this is complicated. 97 00:05:00,860 --> 00:05:03,750 This is difficult, maybe some type of trig function. 98 00:05:03,750 --> 00:05:05,110 But you'll see that it's kind of just the 99 00:05:05,110 --> 00:05:06,740 reverse chain rule. 100 00:05:06,740 --> 00:05:09,860 We have a function here, 1 plus v squared, 101 00:05:09,860 --> 00:05:10,500 an expression here. 102 00:05:10,500 --> 00:05:12,889 And we have its derivative sitting right there. 103 00:05:12,889 --> 00:05:17,039 So the antiderivative of this, and you can make a 104 00:05:17,040 --> 00:05:18,069 substitution if you like. 105 00:05:18,069 --> 00:05:21,790 You could say u is equal to 1 plus v squared, then du is 106 00:05:21,790 --> 00:05:23,240 equal to 2v dv. 107 00:05:23,240 --> 00:05:25,360 And then, well, you would end up saying that the 108 00:05:25,360 --> 00:05:27,319 antiderivative is just the natural log of u. 109 00:05:27,319 --> 00:05:30,759 Or, in this case, the antiderivative of this is just 110 00:05:30,759 --> 00:05:35,639 the natural log of 1 plus v squared. 111 00:05:35,639 --> 00:05:37,439 We don't even have to write an absolute value there. 112 00:05:37,439 --> 00:05:40,939 Because that's always going to be a positive value. 113 00:05:40,939 --> 00:05:44,819 So the natural log of 1 plus v squared. 114 00:05:44,819 --> 00:05:48,259 And I hope I didn't confuse you. 115 00:05:48,259 --> 00:05:49,120 That's how I think about it. 116 00:05:49,120 --> 00:05:51,670 I say, if I have an expression, and I have its 117 00:05:51,670 --> 00:05:54,629 derivative multiplied there, then I can just take the 118 00:05:54,629 --> 00:05:56,860 antiderivative of the whole expression. 119 00:05:56,860 --> 00:05:59,060 And I don't have to worry about what's inside of it. 120 00:05:59,060 --> 00:06:01,980 So if this was a 1 over an x, or 1 over u, it's just the 121 00:06:01,980 --> 00:06:03,270 natural log of it. 122 00:06:03,269 --> 00:06:05,459 So that's how I knew that this was the antiderivative. 123 00:06:05,459 --> 00:06:08,169 And if you don't believe me, use the chain rule and take 124 00:06:08,170 --> 00:06:09,770 the derivative of this, and you'll get this. 125 00:06:09,769 --> 00:06:11,569 And hopefully, it will make a little bit more sense. 126 00:06:11,569 --> 00:06:13,879 But anyway, that's the left hand side, and then that 127 00:06:13,879 --> 00:06:17,000 equals-- Well, this one's easy. 128 00:06:17,000 --> 00:06:21,629 That's the natural log, the absolute value of x. 129 00:06:21,629 --> 00:06:24,180 We could say, plus c, but just so that we can simplify it a 130 00:06:24,180 --> 00:06:27,639 little bit, an arbitrary constant c, we can really just 131 00:06:27,639 --> 00:06:31,099 write that as the natural log of the absolute value of some 132 00:06:31,100 --> 00:06:32,400 constant c. 133 00:06:32,399 --> 00:06:35,399 I mean, this is still some arbitrary constant c. 134 00:06:35,399 --> 00:06:40,879 So we can rewrite this whole equation as the natural log of 135 00:06:40,879 --> 00:06:45,899 1 plus v squared is equal to-- when you add natural logs, you 136 00:06:45,899 --> 00:06:48,239 can essentially just multiply the two numbers that you're 137 00:06:48,240 --> 00:06:51,639 taking the natural log of-- the natural log of, we could 138 00:06:51,639 --> 00:06:54,709 say, the absolute value of cx. 139 00:06:54,709 --> 00:06:56,759 And so the natural log of this is equal to the 140 00:06:56,759 --> 00:07:00,149 natural log of this. 141 00:07:00,149 --> 00:07:07,009 So we could say that 1 plus v squared is equal to cx. 142 00:07:07,009 --> 00:07:08,849 And now we can unsubstitute it. 143 00:07:08,850 --> 00:07:12,700 So we know v is equal to y over x, so let's do that. 144 00:07:12,699 --> 00:07:22,540 So we get 1 plus y over x squared is equal to cx. 145 00:07:22,540 --> 00:07:25,240 Let me scroll this down a little bit. 146 00:07:25,240 --> 00:07:29,100 Let's multiply both sides of the equation times x squared. 147 00:07:29,100 --> 00:07:31,730 We could rewrite this as y squared over x squared. 148 00:07:31,730 --> 00:07:34,900 So we multiply both sides times x squared, you get x 149 00:07:34,899 --> 00:07:41,500 squared plus y squared is equal to cx to the third. 150 00:07:41,500 --> 00:07:42,370 And we're essentially done. 151 00:07:42,370 --> 00:07:45,040 If we want to put all of the variable terms on left hand 152 00:07:45,040 --> 00:07:49,010 side, we could say that this is equal to x squared plus y 153 00:07:49,009 --> 00:07:53,209 squared minus cx to the third is equal to 0. 154 00:07:53,209 --> 00:07:57,199 And this implicitly defined function, or curve, or however 155 00:07:57,199 --> 00:08:00,979 you want to call it, is the solution to our original 156 00:08:00,980 --> 00:08:04,480 homogeneous first order differential equation. 157 00:08:04,480 --> 00:08:06,340 So there you go. 158 00:08:06,339 --> 00:08:07,779 I will see you in the next video. 159 00:08:07,779 --> 00:08:09,339 And now, we're actually going to do something. 160 00:08:09,339 --> 00:08:12,259 We're going to start embarking on higher order 161 00:08:12,259 --> 00:08:12,980 differential equations. 162 00:08:12,980 --> 00:08:15,189 And actually, these are more useful, and in some ways, 163 00:08:15,189 --> 00:08:17,629 easier to do than the homogeneous and the exact 164 00:08:17,629 --> 00:08:19,740 equations that we've been doing so far. 165 00:08:19,740 --> 00:08:21,720 See you in the next video. 166 00:08:21,720 --> 00:08:21,900