1 00:00:00,339 --> 00:00:01,130 GILBERT STRANG: OK. 2 00:00:01,130 --> 00:00:04,700 This is the start of Laplace transforms. 3 00:00:04,700 --> 00:00:09,980 And that's going to take more than one short video. 4 00:00:09,980 --> 00:00:14,560 But I'll devote this video to first order equations, where 5 00:00:14,560 --> 00:00:18,790 the steps are easy and pretty quick. 6 00:00:18,790 --> 00:00:21,590 Then will come second order equations. 7 00:00:21,590 --> 00:00:24,520 So Laplace transforms starting now. 8 00:00:24,520 --> 00:00:28,590 So let me tell you what-- I use a capital 9 00:00:28,590 --> 00:00:34,410 letter for the Laplace transform of little f, a function of t. 10 00:00:34,410 --> 00:00:38,410 The transform is capital F, a function of s. 11 00:00:38,410 --> 00:00:40,550 And you'll see where s comes in. 12 00:00:40,550 --> 00:00:43,490 Or if it's the solution I'm looking at, 13 00:00:43,490 --> 00:00:46,490 y of t, its transform is naturally 14 00:00:46,490 --> 00:00:48,890 called capital Y of s. 15 00:00:48,890 --> 00:00:53,940 So that's what we want-- we want to find y, and we know f. 16 00:00:53,940 --> 00:00:54,830 OK. 17 00:00:54,830 --> 00:00:56,320 So can I do an example? 18 00:00:56,320 --> 00:00:59,770 Well, first tell you what the Laplace transform is. 19 00:00:59,770 --> 00:01:03,980 Suppose the function is f of t. 20 00:01:03,980 --> 00:01:05,560 Here is the transform. 21 00:01:05,560 --> 00:01:10,930 I multiply by e to the minus st, and I integrate from 0 22 00:01:10,930 --> 00:01:12,080 to infinity. 23 00:01:12,080 --> 00:01:13,380 0 to infinity. 24 00:01:13,380 --> 00:01:14,730 Very important. 25 00:01:14,730 --> 00:01:18,060 The function doesn't start until t equals 0, 26 00:01:18,060 --> 00:01:20,380 but it goes on to t equal infinity. 27 00:01:20,380 --> 00:01:24,810 I integrate, and when I integrate, 28 00:01:24,810 --> 00:01:28,230 t disappears but s is still there. 29 00:01:28,230 --> 00:01:30,440 So I have a function of s. 30 00:01:30,440 --> 00:01:32,430 Well, I have to do an example. 31 00:01:32,430 --> 00:01:37,490 So to find the Laplace transform is to do an integration. 32 00:01:37,490 --> 00:01:42,480 And you won't be surprised that the good functions we know 33 00:01:42,480 --> 00:01:45,670 are the ones where we can do the integration 34 00:01:45,670 --> 00:01:49,890 and discover the transform, and make a little table 35 00:01:49,890 --> 00:01:51,960 of nice transforms. 36 00:01:51,960 --> 00:01:56,820 And the number one function we know is the exponential. 37 00:01:56,820 --> 00:01:59,480 So can I find-- for that function, 38 00:01:59,480 --> 00:02:02,190 I'll compute its transform. 39 00:02:02,190 --> 00:02:03,450 So what do I have to do? 40 00:02:03,450 --> 00:02:06,300 I have to integrate from 0 to infinity-- 41 00:02:06,300 --> 00:02:08,330 you might say 0 to infinity is hard, 42 00:02:08,330 --> 00:02:12,570 but it's actually the best-- of my function, which 43 00:02:12,570 --> 00:02:14,045 is e to the at. 44 00:02:14,045 --> 00:02:21,330 So that's my function times e to the minus st dt. 45 00:02:21,330 --> 00:02:21,830 OK. 46 00:02:21,830 --> 00:02:26,390 I can do that integral, because those combine into e 47 00:02:26,390 --> 00:02:32,420 to the a minus st. I can put those together 48 00:02:32,420 --> 00:02:36,750 into e to the a minus st. I integrate 49 00:02:36,750 --> 00:02:45,780 so I get e to the a minus st divided by a minus s. 50 00:02:45,780 --> 00:02:48,830 That's the integral of that. 51 00:02:48,830 --> 00:02:51,680 Because what I have in here is just that. 52 00:02:51,680 --> 00:02:55,980 To integrate the exponential, I just divide by the exponent 53 00:02:55,980 --> 00:02:57,060 there. 54 00:02:57,060 --> 00:03:02,770 And I have just substitute t equal infinity and t equal 0. 55 00:03:02,770 --> 00:03:08,110 So t equal infinity, starting at 0 to infinity. 56 00:03:08,110 --> 00:03:10,140 OK. 57 00:03:10,140 --> 00:03:12,400 Infinity is the nice one. 58 00:03:12,400 --> 00:03:13,790 It's the easy one. 59 00:03:13,790 --> 00:03:20,850 I will look only at s's that are bigger than a. 60 00:03:20,850 --> 00:03:27,570 s larger than a means that this exponential is decreasing to 0. 61 00:03:27,570 --> 00:03:29,940 It gets to 0 at t equal infinity. 62 00:03:29,940 --> 00:03:36,180 So at t equal infinity, that upper limit of the integral 63 00:03:36,180 --> 00:03:38,720 ends up with a 0. 64 00:03:38,720 --> 00:03:42,510 So I just have to subtract the lower limit. 65 00:03:42,510 --> 00:03:43,450 And look how nice. 66 00:03:43,450 --> 00:03:45,810 Now I put in t equal 0. 67 00:03:45,810 --> 00:03:47,850 Well, then that becomes 1. 68 00:03:47,850 --> 00:03:51,100 And it's a lower limit, so it comes with a minus sign. 69 00:03:51,100 --> 00:03:54,120 So it's just the 1 over, the minus sign 70 00:03:54,120 --> 00:03:57,240 will flip that s minus a. 71 00:03:57,240 --> 00:04:01,380 The most important Laplace transform in the world. 72 00:04:01,380 --> 00:04:05,530 Remember, the function was in to the at. 73 00:04:05,530 --> 00:04:09,420 The transform is a function of s. 74 00:04:09,420 --> 00:04:15,400 The original function depended on t and a parameter a. 75 00:04:15,400 --> 00:04:20,269 The result depends on s and a parameter a. 76 00:04:20,269 --> 00:04:26,950 And an engineer would say, here we have the exponent. 77 00:04:26,950 --> 00:04:28,826 The growth rate is a. 78 00:04:28,826 --> 00:04:32,090 And over in the transform-- so this is the transform, 79 00:04:32,090 --> 00:04:32,590 remember. 80 00:04:32,590 --> 00:04:36,620 This is the transform f of x. 81 00:04:36,620 --> 00:04:40,150 In the transform, I see blow up-- 82 00:04:40,150 --> 00:04:49,580 a pole, that's called a pole-- at s equal a. 83 00:04:49,580 --> 00:04:52,470 1/0 is a pole. 84 00:04:52,470 --> 00:04:54,620 And I'm not surprised. 85 00:04:54,620 --> 00:04:58,060 So the answer is blowing up at s equal a. 86 00:04:58,060 --> 00:04:59,230 Well, of course. 87 00:04:59,230 --> 00:05:05,440 If s equals a, then this is the integral of 1 from 0 88 00:05:05,440 --> 00:05:07,470 to infinity, and it's infinite. 89 00:05:07,470 --> 00:05:10,760 So I'm not surprised to see the pole showing up. 90 00:05:10,760 --> 00:05:14,510 The blow up showing up exactly at the exponent a. 91 00:05:14,510 --> 00:05:18,270 But this is a nice transform. 92 00:05:18,270 --> 00:05:20,570 OK. 93 00:05:20,570 --> 00:05:22,520 I need to do one other-- oh, no. 94 00:05:22,520 --> 00:05:27,540 I could already solve the equation. 95 00:05:27,540 --> 00:05:30,090 So let me start with the equation dy 96 00:05:30,090 --> 00:05:36,785 dt minus ay equal 0. 97 00:05:36,785 --> 00:05:40,350 Oh, well, I can take the Laplace transform of 0 98 00:05:40,350 --> 00:05:41,880 is 0, safe enough. 99 00:05:41,880 --> 00:05:45,290 The Laplace transform of y is capital Y. But what's 100 00:05:45,290 --> 00:05:47,170 the transform of this? 101 00:05:47,170 --> 00:05:50,840 Oh, I have to do one more transform for you. 102 00:05:50,840 --> 00:05:55,450 I'm hoping that the transform of the derivative, dy dt, 103 00:05:55,450 --> 00:05:57,660 connects to the transform of y. 104 00:05:57,660 --> 00:06:04,700 So the transform of this guy is the integral from 0 105 00:06:04,700 --> 00:06:10,320 to infinity of that function, whatever it is, times e 106 00:06:10,320 --> 00:06:13,480 to the minus st dt. 107 00:06:13,480 --> 00:06:14,880 This is the transform. 108 00:06:14,880 --> 00:06:19,320 So this Laplace transform. 109 00:06:19,320 --> 00:06:23,520 Now what can I do with that integral? 110 00:06:23,520 --> 00:06:29,910 This is a step that goes back to the beginning of calculus. 111 00:06:29,910 --> 00:06:32,560 But it's easy to forget. 112 00:06:32,560 --> 00:06:37,490 When you see a derivative there inside that integral, 113 00:06:37,490 --> 00:06:40,850 you think, I could integrate by parts. 114 00:06:40,850 --> 00:06:43,340 I could integrate that term and take 115 00:06:43,340 --> 00:06:45,030 the derivative of that term. 116 00:06:45,030 --> 00:06:47,170 That's what integration by parts does. 117 00:06:47,170 --> 00:06:51,200 It moves the derivative away from that and onto that 118 00:06:51,200 --> 00:06:53,270 where it's no problem. 119 00:06:53,270 --> 00:06:59,440 And do you remember that a minus sign comes in when I do this? 120 00:06:59,440 --> 00:07:02,910 So I have the integral from 0 to infinity of-- now 121 00:07:02,910 --> 00:07:07,570 the derivative is coming off of that, so that's just y of t. 122 00:07:07,570 --> 00:07:09,740 And the derivative is going onto that, 123 00:07:09,740 --> 00:07:15,170 so that's minus se to the minus st dt. 124 00:07:15,170 --> 00:07:16,670 Good. 125 00:07:16,670 --> 00:07:19,340 And then do you remember in integration by parts, 126 00:07:19,340 --> 00:07:26,930 there's also another term that comes from y times e 127 00:07:26,930 --> 00:07:29,080 to the minus st? 128 00:07:29,080 --> 00:07:37,250 This is ye to the minus st at 0 and infinity. 129 00:07:37,250 --> 00:07:38,440 OK. 130 00:07:38,440 --> 00:07:40,520 I've integrated by parts. 131 00:07:40,520 --> 00:07:44,390 A very useful, powerful thing, not just a trick. 132 00:07:44,390 --> 00:07:44,960 OK. 133 00:07:44,960 --> 00:07:47,570 Now, can I recognize some of this? 134 00:07:47,570 --> 00:07:53,590 That is minus minus, no problem. 135 00:07:53,590 --> 00:07:55,690 I bring out-- that s is a constant. 136 00:07:55,690 --> 00:07:58,146 Bring it out, s. 137 00:07:58,146 --> 00:08:00,980 Now, what do I have left when I bring out that s? 138 00:08:00,980 --> 00:08:04,360 I have the integral of ye to the minus st dt. 139 00:08:04,360 --> 00:08:08,830 That is exactly the Laplace transform of y. 140 00:08:08,830 --> 00:08:16,370 It's exactly capital Y. 141 00:08:16,370 --> 00:08:18,030 Put the equal sign here. 142 00:08:18,030 --> 00:08:22,740 I'll make that 0 a little smaller, get it out of the way. 143 00:08:22,740 --> 00:08:24,180 OK. sY of s. 144 00:08:24,180 --> 00:08:27,630 So that whole term has a nice form. 145 00:08:27,630 --> 00:08:31,220 When you take the derivative of a function, 146 00:08:31,220 --> 00:08:35,230 you multiply its Laplace transform by s. 147 00:08:35,230 --> 00:08:36,789 That's the rule. 148 00:08:36,789 --> 00:08:38,820 Take the derivative of the function, 149 00:08:38,820 --> 00:08:40,909 multiply the Laplace transform by s. 150 00:08:40,909 --> 00:08:44,550 If we have two derivatives, we'll multiply by s twice. 151 00:08:44,550 --> 00:08:45,510 Easy. 152 00:08:45,510 --> 00:08:48,710 That's why the Laplace transform works. 153 00:08:48,710 --> 00:08:51,220 But now, here is a final term. 154 00:08:51,220 --> 00:08:55,300 y at infinity-- well, and e to the minus 155 00:08:55,300 --> 00:08:58,500 st at t equal infinity, 0. 156 00:08:58,500 --> 00:08:59,600 Forget it. 157 00:08:59,600 --> 00:09:06,940 So I just have to subtract off y at 0 times e to the minus st 158 00:09:06,940 --> 00:09:10,310 at 0, which is 1. 159 00:09:10,310 --> 00:09:12,180 e to the 0 is 1. 160 00:09:12,180 --> 00:09:15,300 So do you see that the initial condition 161 00:09:15,300 --> 00:09:16,670 comes into the transform? 162 00:09:16,670 --> 00:09:18,380 It's like, great. 163 00:09:18,380 --> 00:09:24,560 We have the transform of Y. Now, all this is the transform. 164 00:09:24,560 --> 00:09:33,890 This is the transform of dy dt that we found. 165 00:09:33,890 --> 00:09:35,740 Now, why did I want that? 166 00:09:35,740 --> 00:09:39,390 Because I plan to take the transform 167 00:09:39,390 --> 00:09:42,910 of every term in my equation. 168 00:09:42,910 --> 00:09:47,480 So like there are two steps to using the Laplace transform. 169 00:09:47,480 --> 00:09:53,370 One is to compute some transforms like this one, 170 00:09:53,370 --> 00:09:56,680 and some rules like this one. 171 00:09:56,680 --> 00:09:58,660 That's the preparation step. 172 00:09:58,660 --> 00:10:01,640 That comes from just looking at these integrals. 173 00:10:01,640 --> 00:10:06,360 And then to use them, I'm going to take the Laplace 174 00:10:06,360 --> 00:10:08,600 transform of every term. 175 00:10:08,600 --> 00:10:10,300 So I have an equation. 176 00:10:10,300 --> 00:10:12,680 I take the Laplace transform of every term. 177 00:10:12,680 --> 00:10:14,390 I've got another equation. 178 00:10:14,390 --> 00:10:23,440 So the Laplace transform of this is sY of s minus y of 0. 179 00:10:23,440 --> 00:10:26,080 That was a Laplace transform of this part. 180 00:10:26,080 --> 00:10:29,610 Now the Laplace transform of this is minus a, 181 00:10:29,610 --> 00:10:32,840 a constant, Y of x. 182 00:10:32,840 --> 00:10:36,250 And the Laplace transform of 0 is 0. 183 00:10:36,250 --> 00:10:38,980 Do you realize what we've done? 184 00:10:38,980 --> 00:10:41,470 I've taken a differential equation 185 00:10:41,470 --> 00:10:44,450 and I've produced an algebra equation. 186 00:10:44,450 --> 00:10:47,620 That's the point of the Laplace transform, 187 00:10:47,620 --> 00:10:51,500 to turn differential equations-- derivatives turn 188 00:10:51,500 --> 00:10:54,740 into multiplications, algebra. 189 00:10:54,740 --> 00:10:57,810 So all the terms turn into that one. 190 00:10:57,810 --> 00:11:02,110 And now comes-- so that's big step one. 191 00:11:02,110 --> 00:11:03,690 Transform every term. 192 00:11:03,690 --> 00:11:07,000 Get an algebra problem for each s. 193 00:11:07,000 --> 00:11:11,840 We've changed from t, time in the differential equation, 194 00:11:11,840 --> 00:11:15,110 to s in Laplace transform. 195 00:11:15,110 --> 00:11:17,340 Now solve this. 196 00:11:17,340 --> 00:11:18,770 So how am I going to solve that? 197 00:11:18,770 --> 00:11:22,080 I'm going to put y of 0 on to the right-hand side. 198 00:11:27,000 --> 00:11:30,180 And then I have Y of s times s minus a. 199 00:11:30,180 --> 00:11:34,615 So I will divide by s minus a. 200 00:11:34,615 --> 00:11:38,380 And that gives me Y of s. 201 00:11:38,380 --> 00:11:40,130 So that was easy to do. 202 00:11:40,130 --> 00:11:42,550 The algebra problem was easy to solve. 203 00:11:42,550 --> 00:11:45,110 The differential equation more serious. 204 00:11:45,110 --> 00:11:45,610 OK. 205 00:11:45,610 --> 00:11:48,699 The algebra problem is easy. 206 00:11:48,699 --> 00:11:49,365 Are we finished? 207 00:11:53,200 --> 00:11:59,230 Got the answer, but we're in the s variable, the s domain. 208 00:11:59,230 --> 00:12:01,920 I've got to get back to-- so now this 209 00:12:01,920 --> 00:12:10,340 is going to be an inverse Laplace transform. 210 00:12:10,340 --> 00:12:12,190 That's the inverse transform. 211 00:12:12,190 --> 00:12:17,800 To give me back y of t equals what? 212 00:12:17,800 --> 00:12:20,090 How am I going to do the inverse transform? 213 00:12:20,090 --> 00:12:25,190 So now I have the transform of the answer, 214 00:12:25,190 --> 00:12:26,930 and I want the answer. 215 00:12:26,930 --> 00:12:32,530 I have to invert that transform and get out of s and back to t. 216 00:12:32,530 --> 00:12:36,500 Well, y of 0 is a constant. 217 00:12:36,500 --> 00:12:39,080 Laplace transform is linear, no problem. 218 00:12:39,080 --> 00:12:41,990 So have y of 0 from that. 219 00:12:41,990 --> 00:12:45,750 And now I have 1 over s minus a. 220 00:12:45,750 --> 00:12:51,130 So I'm asking myself, what is the function whose transform is 221 00:12:51,130 --> 00:12:53,430 1 over s minus a? 222 00:12:53,430 --> 00:12:56,490 Then it's that function that I want to put in there. 223 00:12:56,490 --> 00:13:00,260 And what is the function whose transform is 1 over s minus a? 224 00:13:00,260 --> 00:13:01,880 It's the one we did. 225 00:13:01,880 --> 00:13:04,320 It's this one up here. 226 00:13:04,320 --> 00:13:09,140 1 over s minus a came from the function e to the at. 227 00:13:09,140 --> 00:13:13,840 So that 1 over s minus a, when I transform back, 228 00:13:13,840 --> 00:13:16,310 is the e to the at. 229 00:13:16,310 --> 00:13:18,460 And I'm golden. 230 00:13:18,460 --> 00:13:24,370 And that you recognize, of course, as the correct answer, 231 00:13:24,370 --> 00:13:29,460 correct solution to this differential equation. 232 00:13:29,460 --> 00:13:36,170 The initial value y of 0 takes off with exponential e 233 00:13:36,170 --> 00:13:36,800 to the at. 234 00:13:36,800 --> 00:13:38,210 No problem. 235 00:13:38,210 --> 00:13:39,390 OK. 236 00:13:39,390 --> 00:13:44,290 Can I do one more example of a first order equation? 237 00:13:44,290 --> 00:13:47,630 Now I'm going to put it in an f of t. 238 00:13:47,630 --> 00:13:49,770 I'm going to put in a source term . 239 00:13:49,770 --> 00:13:54,310 So I'll do all the same stuff, but I'm 240 00:13:54,310 --> 00:13:56,460 going to have an f of t. 241 00:13:56,460 --> 00:14:00,680 And what shall I take for-- I'll take an exponential again, e 242 00:14:00,680 --> 00:14:03,970 to the ct. 243 00:14:03,970 --> 00:14:06,660 So that's my right-hand side. 244 00:14:06,660 --> 00:14:13,740 Can I do the same idea, the central idea? 245 00:14:13,740 --> 00:14:19,280 Take my differential equation, transform every term. 246 00:14:19,280 --> 00:14:21,810 I've started with a time equation 247 00:14:21,810 --> 00:14:24,190 and I'm going to get an s equation. 248 00:14:24,190 --> 00:14:29,470 So again, dy dt minus ay, that transformed to-- what did that 249 00:14:29,470 --> 00:14:30,890 transform to? 250 00:14:30,890 --> 00:14:36,510 sY of s minus y of 0. 251 00:14:36,510 --> 00:14:37,880 Came from there. 252 00:14:37,880 --> 00:14:41,885 Minus aY of 0-- minus aY of s. 253 00:14:45,570 --> 00:14:48,650 Minus aY of s. 254 00:14:48,650 --> 00:14:51,620 And on the right-hand side, I have the transform of e 255 00:14:51,620 --> 00:14:52,480 to the ct. 256 00:14:52,480 --> 00:14:54,650 We're getting good at this transform. 257 00:14:54,650 --> 00:15:00,231 1 over s minus c, instead of a at c. 258 00:15:00,231 --> 00:15:00,730 OK. 259 00:15:00,730 --> 00:15:03,820 That's our equation transform. 260 00:15:03,820 --> 00:15:06,080 Now algebra. 261 00:15:06,080 --> 00:15:09,860 I just pull Y of s out of that. 262 00:15:09,860 --> 00:15:12,560 How am I going to pull Y of s out of this equation? 263 00:15:12,560 --> 00:15:14,975 Well, I'll move y of 0 to the other side. 264 00:15:18,770 --> 00:15:21,250 And I'll divide by s minus a. 265 00:15:21,250 --> 00:15:22,710 Look at it. 266 00:15:22,710 --> 00:15:26,960 Y of s is-- OK. 267 00:15:26,960 --> 00:15:31,320 I have 1 over s minus c. 268 00:15:31,320 --> 00:15:34,170 And I have an s minus a that I'm dividing by. 269 00:15:34,170 --> 00:15:37,790 S minus a. 270 00:15:37,790 --> 00:15:43,310 And then I have the y of 0 over s minus a. 271 00:15:47,910 --> 00:15:52,380 I've transformed the differential equation 272 00:15:52,380 --> 00:15:54,210 to an s equation. 273 00:15:54,210 --> 00:15:58,160 I've just done simple algebra to solve that equation. 274 00:15:58,160 --> 00:16:00,020 And I've got two terms. 275 00:16:00,020 --> 00:16:02,780 Two terms. 276 00:16:02,780 --> 00:16:03,930 You see that term? 277 00:16:03,930 --> 00:16:05,900 That's what I had before. 278 00:16:05,900 --> 00:16:07,650 That's what I had just there. 279 00:16:07,650 --> 00:16:10,580 The inverse transform was this. 280 00:16:10,580 --> 00:16:11,550 No problem. 281 00:16:11,550 --> 00:16:17,780 That's the null solution that's coming out of the initial value 282 00:16:17,780 --> 00:16:18,460 . 283 00:16:18,460 --> 00:16:24,180 The new term that's coming from the e to the ct, 284 00:16:24,180 --> 00:16:28,030 coming from the force, is this one. 285 00:16:28,030 --> 00:16:33,320 And I have to do its inverse transform. 286 00:16:33,320 --> 00:16:38,020 I have to figure out what function has that transform. 287 00:16:38,020 --> 00:16:40,890 And you may say, that's completely new. 288 00:16:40,890 --> 00:16:44,580 But we can connect it to the one we know. 289 00:16:44,580 --> 00:16:45,080 OK. 290 00:16:45,080 --> 00:16:54,340 So that will give me the same inverse transform, the growing 291 00:16:54,340 --> 00:16:55,490 exponential. 292 00:16:55,490 --> 00:16:58,360 But what does this one give? 293 00:16:58,360 --> 00:17:01,190 That's a key question. 294 00:17:01,190 --> 00:17:07,200 We have to be able to do-- invert, figure out what 295 00:17:07,200 --> 00:17:09,800 function has that transform? 296 00:17:09,800 --> 00:17:14,810 The function will involve a and c and t, the time. 297 00:17:14,810 --> 00:17:21,260 And s, the transform variable, will become t, the time period. 298 00:17:21,260 --> 00:17:23,170 So that's the big question. 299 00:17:23,170 --> 00:17:27,440 What do I do with this? 300 00:17:27,440 --> 00:17:29,840 And notice, it has two poles. 301 00:17:29,840 --> 00:17:34,390 It blows up at s equal a, and it blows up at s equal c. 302 00:17:34,390 --> 00:17:43,020 And I have to figure out-- well, actually, by good luck, 303 00:17:43,020 --> 00:17:46,030 I want to separate those two poles. 304 00:17:46,030 --> 00:17:47,540 Because if I separate the two poles, 305 00:17:47,540 --> 00:17:51,430 I know what to do with a blow up at s equal a and a blow up at s 306 00:17:51,430 --> 00:17:52,210 equal c. 307 00:17:52,210 --> 00:17:56,190 The problem is, right now I have both blow ups at once. 308 00:17:56,190 --> 00:17:57,800 So I'm going to separate that. 309 00:17:57,800 --> 00:18:01,480 And that's called partial fractions. 310 00:18:01,480 --> 00:18:05,150 So I will have to say more about partial fractions. 311 00:18:05,150 --> 00:18:07,580 Right now, let me just do it. 312 00:18:07,580 --> 00:18:10,860 That expression there, I'll take this guy away. 313 00:18:10,860 --> 00:18:14,330 Because it gives that term that we know. 314 00:18:18,520 --> 00:18:20,610 It's this one. 315 00:18:20,610 --> 00:18:21,910 Is that one. 316 00:18:21,910 --> 00:18:25,420 It's this two poles thing that I want 317 00:18:25,420 --> 00:18:28,430 to separate those two poles. 318 00:18:28,430 --> 00:18:30,450 So this is algebra again. 319 00:18:30,450 --> 00:18:32,400 Partial fractions is just algebra. 320 00:18:32,400 --> 00:18:33,900 No calculus. 321 00:18:33,900 --> 00:18:36,080 No derivatives are in here. 322 00:18:36,080 --> 00:18:42,190 I just want to write that as 1 over s minus c. 323 00:18:42,190 --> 00:18:44,250 It turns out-- look. 324 00:18:44,250 --> 00:18:47,020 There's a way to remember the answer. 325 00:18:47,020 --> 00:19:00,730 s minus c times c minus a and 1 over s minus a. 326 00:19:00,730 --> 00:19:03,700 And now a minus c. 327 00:19:06,448 --> 00:19:11,070 Do you see that that has only one pole at s equals c? 328 00:19:11,070 --> 00:19:12,870 This is just a number. 329 00:19:12,870 --> 00:19:15,260 This has one pole at s equals a. 330 00:19:15,260 --> 00:19:16,250 That's just a number. 331 00:19:16,250 --> 00:19:19,660 In fact, those numbers are the opposite. 332 00:19:19,660 --> 00:19:24,150 So now, are we golden? 333 00:19:24,150 --> 00:19:27,910 I can take the inverse transform with just one pole. 334 00:19:27,910 --> 00:19:35,050 So now that gives me the solution y from-- so this 335 00:19:35,050 --> 00:19:40,310 is just a constant, 1 over c minus a. 336 00:19:40,310 --> 00:19:44,720 And what is the inverse transform of this? 337 00:19:44,720 --> 00:19:49,230 That's the simple pole at c. 338 00:19:49,230 --> 00:19:52,805 It came from a pure exponential, e to the ct. 339 00:19:56,350 --> 00:19:57,375 Right? 340 00:19:57,375 --> 00:20:01,070 And now this guy, this one. 341 00:20:01,070 --> 00:20:01,970 OK. 342 00:20:01,970 --> 00:20:05,730 Well, this has a minus c, which is the opposite of c minus a. 343 00:20:05,730 --> 00:20:10,410 So if I put a minus sign, I can put them all over c minus a. 344 00:20:10,410 --> 00:20:11,090 Look at that. 345 00:20:11,090 --> 00:20:12,160 Look at this. 346 00:20:12,160 --> 00:20:15,280 C minus a is in both of these. 347 00:20:15,280 --> 00:20:17,570 Here it is with a plus sign. 348 00:20:17,570 --> 00:20:24,390 And that transform came from that function. 349 00:20:24,390 --> 00:20:26,550 Here it is with a minus sign. 350 00:20:26,550 --> 00:20:28,850 So I want a minus there. 351 00:20:28,850 --> 00:20:34,180 And what function gave me that transform? 352 00:20:34,180 --> 00:20:36,780 e to the at, right? 353 00:20:36,780 --> 00:20:39,040 That's the one we know. 354 00:20:39,040 --> 00:20:44,680 The unforgettable transform of a simple exponential, 355 00:20:44,680 --> 00:20:47,660 e to the at. 356 00:20:47,660 --> 00:20:53,360 That is the particular solution. 357 00:20:53,360 --> 00:21:01,440 So the Laplace transform, we transformed the differential 358 00:21:01,440 --> 00:21:02,980 equation. 359 00:21:02,980 --> 00:21:04,900 We got an algebra equation. 360 00:21:04,900 --> 00:21:08,060 We solved that algebra equation, and then we 361 00:21:08,060 --> 00:21:15,240 had to go backwards to find what function had this transform y. 362 00:21:15,240 --> 00:21:19,680 And to see that, clearly we had to use this partial fraction 363 00:21:19,680 --> 00:21:23,980 idea which separated these two poles into one pole 364 00:21:23,980 --> 00:21:29,430 there, when s is c, and another pole, when s is a. 365 00:21:29,430 --> 00:21:31,520 We've got two easy fractions. 366 00:21:31,520 --> 00:21:36,110 The easy fractions each gave me an exponential. 367 00:21:36,110 --> 00:21:38,970 And the final result was this one. 368 00:21:38,970 --> 00:21:41,380 And I don't know if you remember that. 369 00:21:41,380 --> 00:21:47,670 That is the correct solution to the first order linear 370 00:21:47,670 --> 00:21:51,100 constant coefficient equation, the simple equation 371 00:21:51,100 --> 00:21:56,070 there, when the right-hand side is e to the ct. 372 00:21:56,070 --> 00:21:59,480 So our final solution then is the null solution 373 00:21:59,480 --> 00:22:02,620 with the initial value in it. 374 00:22:02,620 --> 00:22:08,280 And that function comes from the right-hand side, comes 375 00:22:08,280 --> 00:22:11,500 from the force, e to the ct. 376 00:22:11,500 --> 00:22:16,210 And so that's how Laplace transforms work. 377 00:22:16,210 --> 00:22:19,490 Take the Laplace transform of every term. 378 00:22:19,490 --> 00:22:25,421 Solve for y of s, and try your best to invert that transform. 379 00:22:25,421 --> 00:22:25,920 OK. 380 00:22:25,920 --> 00:22:30,830 More of that coming in the next lecture on Laplace transforms. 381 00:22:30,830 --> 00:22:32,630 Thank you.