1 00:00:00,000 --> 00:00:00,670 2 00:00:00,670 --> 00:00:03,339 I've been doing a ton of videos on the mechanics of 3 00:00:03,339 --> 00:00:06,529 taking the Laplace Transform, but you've been sitting 4 00:00:06,530 --> 00:00:09,620 through them always wondering, what am I learning this for? 5 00:00:09,619 --> 00:00:12,669 And now I'll show you, at least in the context of 6 00:00:12,669 --> 00:00:13,869 differential equations. 7 00:00:13,869 --> 00:00:16,140 And I've gotten a bunch of letters 8 00:00:16,140 --> 00:00:17,679 on the Laplace Transform. 9 00:00:17,679 --> 00:00:18,570 What does it really mean? 10 00:00:18,570 --> 00:00:19,390 And and all that. 11 00:00:19,390 --> 00:00:21,050 And those are excellent questions and you should 12 00:00:21,050 --> 00:00:22,120 strive for that. 13 00:00:22,120 --> 00:00:25,410 It's hard to really have an intuition of the Laplace 14 00:00:25,410 --> 00:00:28,030 Transform in the differential equations context, other than 15 00:00:28,030 --> 00:00:32,630 it being a very useful tool that converts differential or 16 00:00:32,630 --> 00:00:36,530 integral problems into algebra problems. But I'll give you a 17 00:00:36,530 --> 00:00:39,980 hint, and if you want a path to learn it in, you should 18 00:00:39,979 --> 00:00:42,519 learn about Fourier series and Fourier Transforms, which are 19 00:00:42,520 --> 00:00:44,710 very similar to Laplace Transforms. And that'll 20 00:00:44,710 --> 00:00:48,359 actually build up the intuition on what the 21 00:00:48,359 --> 00:00:50,609 frequency domain is all about. 22 00:00:50,609 --> 00:00:53,200 Well anyway, let's actually use the Laplace Transform to 23 00:00:53,200 --> 00:00:54,630 solve a differential equation. 24 00:00:54,630 --> 00:00:56,679 And this is one we've seen before. 25 00:00:56,679 --> 00:00:59,210 So let me see. 26 00:00:59,210 --> 00:01:03,299 So let's say the differential equation is y prime prime, 27 00:01:03,299 --> 00:01:07,329 plus 5, times the first derivative, plus 28 00:01:07,329 --> 00:01:09,209 6y, is equal to 0. 29 00:01:09,209 --> 00:01:10,929 And you know how to solve this one, but I just want to show 30 00:01:10,930 --> 00:01:13,490 you, with a fairly straightforward differential 31 00:01:13,489 --> 00:01:15,539 equation, that you could solve it with the Laplace Transform. 32 00:01:15,540 --> 00:01:17,930 And actually, you end up having a 33 00:01:17,930 --> 00:01:19,630 characteristic equation. 34 00:01:19,629 --> 00:01:27,399 And the initial conditions are y of 0 is equal to 2, and y 35 00:01:27,400 --> 00:01:32,490 prime of 0 is equal to 3. 36 00:01:32,489 --> 00:01:36,099 Now, to use the Laplace Transform here, we essentially 37 00:01:36,099 --> 00:01:38,280 just take the Laplace Transform of both sides of 38 00:01:38,280 --> 00:01:40,460 this equation. 39 00:01:40,459 --> 00:01:42,589 Let me use a more vibrant color. 40 00:01:42,590 --> 00:01:47,060 So we get the Laplace Transform of y the second 41 00:01:47,060 --> 00:01:51,090 derivative, plus-- well we could say the Laplace 42 00:01:51,090 --> 00:01:53,590 Transform of 5 times y prime, but that's the same thing as 5 43 00:01:53,590 --> 00:01:56,745 times the Laplace Transform-- y prime. 44 00:01:56,745 --> 00:01:59,310 45 00:01:59,310 --> 00:02:06,189 y prime plus 6 times the Laplace Transform of y. 46 00:02:06,189 --> 00:02:06,909 And let me ask you a question. 47 00:02:06,909 --> 00:02:09,780 What's the Laplace Transform of 0? 48 00:02:09,780 --> 00:02:11,129 Let me do that. 49 00:02:11,129 --> 00:02:12,990 So the Laplace Transform of 0 would be be the integral from 50 00:02:12,990 --> 00:02:19,159 0 to infinity, of 0 times e to the minus stdt. 51 00:02:19,159 --> 00:02:20,329 So this is a 0 in here. 52 00:02:20,330 --> 00:02:21,400 So this is equal to 0. 53 00:02:21,400 --> 00:02:23,340 So the Laplace Transform of 0 is 0. 54 00:02:23,340 --> 00:02:25,870 And that's good, because I didn't have space to do 55 00:02:25,870 --> 00:02:28,560 another curly L. 56 00:02:28,560 --> 00:02:30,729 So what are the Laplace Transforms of these things? 57 00:02:30,729 --> 00:02:33,209 Well this is where we break out one of the useful 58 00:02:33,210 --> 00:02:36,500 properties that we learned. 59 00:02:36,500 --> 00:02:37,530 Let me write it over here. 60 00:02:37,530 --> 00:02:40,819 I think that's going to need as much 61 00:02:40,819 --> 00:02:42,656 real estate as possible. 62 00:02:42,656 --> 00:02:44,319 Let me erase this. 63 00:02:44,319 --> 00:02:48,849 64 00:02:48,849 --> 00:02:52,079 So we learned that the Laplace Transform-- I'll do it here. 65 00:02:52,080 --> 00:02:54,860 Actually, I'll do it down here. 66 00:02:54,860 --> 00:03:01,150 The Laplace Transform of f prime, or we could even say y 67 00:03:01,150 --> 00:03:08,990 prime, is equal to s times the Laplace Transform of 68 00:03:08,990 --> 00:03:13,689 y, minus y of 0. 69 00:03:13,689 --> 00:03:15,800 We proved that to you. 70 00:03:15,800 --> 00:03:19,300 And this is extremely important to know. 71 00:03:19,300 --> 00:03:20,800 So let's see if we can apply that. 72 00:03:20,800 --> 00:03:23,810 So the Laplace Transform of y prime prime, if we apply that, 73 00:03:23,810 --> 00:03:33,295 that's equal to s times the Laplace Transform of-- well if 74 00:03:33,295 --> 00:03:35,539 we go from y prime to y, you're just taking the 75 00:03:35,539 --> 00:03:37,389 anti-derivative, so if you're taking the anti-derivative of 76 00:03:37,389 --> 00:03:40,139 y, of the second derivative, we just end up with the first 77 00:03:40,139 --> 00:03:45,949 derivative-- minus the first derivative at 0. 78 00:03:45,949 --> 00:03:48,530 Notice, we're already using our initial conditions. 79 00:03:48,530 --> 00:03:51,129 I won't substitute it just yet. 80 00:03:51,129 --> 00:03:56,949 And then we end up with plus 5, times-- I'll write it every 81 00:03:56,949 --> 00:04:07,530 time-- so plus 5 times the Laplace Transform of y prime, 82 00:04:07,530 --> 00:04:12,800 plus 6 times the Laplace Transform of y. 83 00:04:12,800 --> 00:04:13,969 All of that is equal to 0. 84 00:04:13,969 --> 00:04:20,490 So just to be clear, all I did is I expanded this into this 85 00:04:20,490 --> 00:04:22,350 using this. 86 00:04:22,350 --> 00:04:25,410 So how can we rewrite the Laplace Transform of y prime? 87 00:04:25,410 --> 00:04:27,870 Well, we could use this once again, so let's do that. 88 00:04:27,870 --> 00:04:31,629 So this over here-- I'll do it in magenta-- this is equal to 89 00:04:31,629 --> 00:04:34,829 s times what? 90 00:04:34,829 --> 00:04:38,949 s times the Laplace Transform of y prime. 91 00:04:38,949 --> 00:04:48,409 Well that's s times the Laplace Transform of y, minus 92 00:04:48,410 --> 00:04:52,220 y of 0, right? 93 00:04:52,220 --> 00:04:55,300 I took this part and replaced it with what I have in 94 00:04:55,300 --> 00:04:56,310 parentheses. 95 00:04:56,310 --> 00:05:05,800 So minus y prime of 0-- and now I'll switch colors-- plus 96 00:05:05,800 --> 00:05:09,370 5 times-- once again the Laplace Transform of y prime. 97 00:05:09,370 --> 00:05:10,360 Well we can use this again. 98 00:05:10,360 --> 00:05:24,550 So 5 times s times Laplace Transform of y, minus y of 0, 99 00:05:24,550 --> 00:05:27,540 plus 6 times the Laplace Transform-- oh I ran out of 100 00:05:27,540 --> 00:05:31,160 space, I'll do it in another line-- plus 6 times the 101 00:05:31,160 --> 00:05:34,250 Laplace Transform of y. 102 00:05:34,250 --> 00:05:35,189 All of that is equal to 0. 103 00:05:35,189 --> 00:05:36,730 I know this looks really confusing but we'll 104 00:05:36,730 --> 00:05:37,819 simplify right now. 105 00:05:37,819 --> 00:05:39,680 And we could get rid of this right here, because we've used 106 00:05:39,680 --> 00:05:42,550 it as much as we need to. 107 00:05:42,550 --> 00:05:45,050 So now we just simplify. 108 00:05:45,050 --> 00:05:46,900 And notice, using the Laplace Transform, we didn't have to 109 00:05:46,899 --> 00:05:49,029 guess at a general solution or anything like that. 110 00:05:49,029 --> 00:05:50,899 Even when we did a characteristic equation, we 111 00:05:50,899 --> 00:05:53,029 guessed what the original general solution was. 112 00:05:53,029 --> 00:05:54,899 Now we're just taking Laplace Transforms, and let's see 113 00:05:54,899 --> 00:05:57,594 where this gets us. 114 00:05:57,595 --> 00:05:59,410 And actually I just want to make clear, because I know 115 00:05:59,410 --> 00:06:09,830 it's very confusing, so I rewrote this part as this. 116 00:06:09,829 --> 00:06:14,259 And I rewrote this thing as this. 117 00:06:14,259 --> 00:06:15,360 And everything else is the same. 118 00:06:15,360 --> 00:06:17,610 But now let's simplify the math. 119 00:06:17,610 --> 00:06:23,930 So we get s squared, times the Laplace Transform of y-- I'm 120 00:06:23,930 --> 00:06:28,900 going to write smaller, I've learned my lesson-- minus s 121 00:06:28,899 --> 00:06:32,149 times y of 0. 122 00:06:32,149 --> 00:06:35,859 Let's substitute y of 0 here. y of 0 is 2, so s times y of 0 123 00:06:35,860 --> 00:06:41,970 is 2 times s, so 2s, distribute that s, minus y 124 00:06:41,970 --> 00:06:42,770 prime of 0. 125 00:06:42,769 --> 00:06:44,589 Y prime of 0 is 3. 126 00:06:44,589 --> 00:06:52,719 So minus 3, plus-- so we have 5 times s times the Laplace 127 00:06:52,720 --> 00:06:58,830 Transform of y, so plus 5s times the Laplace Transform of 128 00:06:58,829 --> 00:07:07,449 y, minus 5 times y of 0. y of 0 is 2, so minus 10. 129 00:07:07,449 --> 00:07:08,979 Minus 10, right? 130 00:07:08,980 --> 00:07:14,040 5 times-- this is 2 right here-- so 5 times 2, plus 6 131 00:07:14,040 --> 00:07:18,950 times the Laplace Transform of y. 132 00:07:18,949 --> 00:07:22,449 All of that is equal to 0. 133 00:07:22,449 --> 00:07:26,069 Now, let's group our Laplace Transform of y terms and our 134 00:07:26,069 --> 00:07:28,800 constant terms, and we should be hopefully 135 00:07:28,800 --> 00:07:30,790 getting some place. 136 00:07:30,790 --> 00:07:33,750 So let's see, my Laplace Transform of y terms, I have 137 00:07:33,750 --> 00:07:39,699 this one, I have this one, and I have that one. 138 00:07:39,699 --> 00:07:40,479 So what am I left with? 139 00:07:40,480 --> 00:07:43,020 Well let me factor out the Laplace Transform of y part. 140 00:07:43,019 --> 00:07:47,159 So I get the Laplace Transform of y-- and that's good because 141 00:07:47,160 --> 00:07:50,980 it's a pain to keep writing it over and over-- times s 142 00:07:50,980 --> 00:08:02,120 squared plus 5s plus 6. 143 00:08:02,120 --> 00:08:05,370 So those are all my Laplace Transform terms. And then I 144 00:08:05,370 --> 00:08:09,459 have my constant terms. So let's see, I have 1s, so minus 145 00:08:09,459 --> 00:08:20,039 2s, minus 3, minus 10, is equal to 0. 146 00:08:20,040 --> 00:08:21,480 And what can we do here? 147 00:08:21,480 --> 00:08:23,480 Well, this is interesting, first of all. 148 00:08:23,480 --> 00:08:28,270 Notice that the coefficients on the Laplace Transform of y 149 00:08:28,269 --> 00:08:31,199 terms, that those are that characteristic equation that 150 00:08:31,199 --> 00:08:33,840 we dealt with so much, and that is hopefully, to some 151 00:08:33,840 --> 00:08:34,908 degree, second nature to you. 152 00:08:34,908 --> 00:08:38,240 So that's a little bit of a clue, and if you want some 153 00:08:38,240 --> 00:08:40,690 very tenuous connections, well that makes a lot of sense. 154 00:08:40,690 --> 00:08:43,030 Because the characteristic equation to get that, we 155 00:08:43,029 --> 00:08:47,029 substituted e to the rt, and the Laplace Transform involves 156 00:08:47,029 --> 00:08:48,139 very similar function. 157 00:08:48,139 --> 00:08:49,669 But anyway, let's go back to the problem. 158 00:08:49,669 --> 00:08:51,399 So how do we solve this? 159 00:08:51,399 --> 00:08:53,189 And actually, let me just give you the big picture here, 160 00:08:53,190 --> 00:08:54,630 because this is a good point. 161 00:08:54,629 --> 00:08:56,830 What I'm going to do is I'm going to solve this. 162 00:08:56,830 --> 00:08:58,280 I'm going to say the Laplace Transform of 163 00:08:58,279 --> 00:09:00,970 y is equal to something. 164 00:09:00,970 --> 00:09:03,259 And then I'm going to say, boy, what functions the 165 00:09:03,259 --> 00:09:05,100 Laplace Transform is at something? 166 00:09:05,100 --> 00:09:06,220 And then I'll have the solution. 167 00:09:06,220 --> 00:09:09,009 If that confuses you, just wait and hopefully it'll make 168 00:09:09,009 --> 00:09:09,379 some sense. 169 00:09:09,379 --> 00:09:11,629 From here until that point it's just some 170 00:09:11,629 --> 00:09:13,610 fairly hairy algebra. 171 00:09:13,610 --> 00:09:16,830 So let's scroll down a little bit, just so we have some 172 00:09:16,830 --> 00:09:17,900 breathing room. 173 00:09:17,899 --> 00:09:23,720 And so I get the Laplace Transform of y, times s 174 00:09:23,720 --> 00:09:29,800 squared, plus 5s, plus 6, is equal to-- let's add these 175 00:09:29,799 --> 00:09:34,949 terms to both sides of this equation-- is equal to 2s plus 176 00:09:34,950 --> 00:09:40,930 3 plus 10-- oh, that's silly-- plus 13. 177 00:09:40,929 --> 00:09:42,629 This is minus 13 here. 178 00:09:42,629 --> 00:09:44,629 A phone call. 179 00:09:44,629 --> 00:09:45,769 Who's calling? 180 00:09:45,769 --> 00:09:48,649 I think it's some kind of marketing phone call. 181 00:09:48,649 --> 00:09:51,600 Anyway, 2s plus 13, and now what can I do? 182 00:09:51,600 --> 00:09:51,909 Well. 183 00:09:51,909 --> 00:09:55,129 Let's divide both sides by this s squared plus 5s plus 6. 184 00:09:55,129 --> 00:10:08,029 So I get the Laplace Transform of y is equal to 2s plus 13, 185 00:10:08,029 --> 00:10:13,909 over s squared plus 5s plus 6. 186 00:10:13,909 --> 00:10:15,539 Now we're almost done. 187 00:10:15,539 --> 00:10:17,439 Everything here is just a little bit of algebra. 188 00:10:17,440 --> 00:10:19,350 So now we're almost done. 189 00:10:19,350 --> 00:10:21,730 We haven't solved for y yet, but we know that the Laplace 190 00:10:21,730 --> 00:10:23,970 Transform of y is equal to this. 191 00:10:23,970 --> 00:10:27,160 Now, if we just had this in our table of our Laplace 192 00:10:27,159 --> 00:10:29,209 Transforms, we would immediately know what y was, 193 00:10:29,210 --> 00:10:31,750 but I don't see something, or I don't remember anything we 194 00:10:31,750 --> 00:10:35,789 did in our table that looks like this expression of s. 195 00:10:35,789 --> 00:10:39,129 I'm essentially out of time, so the next video we're going 196 00:10:39,129 --> 00:10:42,950 to figure out what functions Laplace Transform is this. 197 00:10:42,950 --> 00:10:45,490 And it actually turns out it's a sum of things we already 198 00:10:45,490 --> 00:10:47,590 know, and we just have to manipulate this a little bit 199 00:10:47,590 --> 00:10:48,639 algebraically. 200 00:10:48,639 --> 00:10:50,620 See you in the next video. 201 00:10:50,620 --> 00:10:50,899