1 00:00:00,000 --> 00:00:00,520 2 00:00:00,520 --> 00:00:03,089 Now that you've had a little bit of exposure to what a 3 00:00:03,089 --> 00:00:05,650 convolution is, I can introduce you to the 4 00:00:05,650 --> 00:00:14,900 convolution theorem, or at least in the context of-- 5 00:00:14,900 --> 00:00:17,469 there may be other convolution theorems-- but we're talking 6 00:00:17,469 --> 00:00:20,049 about differential equations and Laplace transforms. So 7 00:00:20,050 --> 00:00:23,440 this is the convolution theorem as applies to Laplace 8 00:00:23,440 --> 00:00:27,400 transforms. And it tells us that if I have a function f of 9 00:00:27,399 --> 00:00:32,259 t-- and I can define its Laplace transform as, let's 10 00:00:32,259 --> 00:00:37,130 see, the Laplace transform of f of t is capital F of s. 11 00:00:37,130 --> 00:00:38,660 We've done that before. 12 00:00:38,659 --> 00:00:43,579 And if I have another function, g of t, and I take 13 00:00:43,579 --> 00:00:50,009 its Laplace transform, that of course is capital G of s. 14 00:00:50,009 --> 00:00:53,219 Then if we were to convolute these two functions, so if I 15 00:00:53,219 --> 00:00:58,670 were to take f and I were to convolute it with g, which is 16 00:00:58,670 --> 00:01:00,990 going to be another function of t-- and 17 00:01:00,990 --> 00:01:02,039 we already saw this. 18 00:01:02,039 --> 00:01:02,969 We saw that in the last video. 19 00:01:02,969 --> 00:01:06,859 I convoluted sine and cosine. 20 00:01:06,859 --> 00:01:09,459 So this is going to be a function of t. 21 00:01:09,459 --> 00:01:13,349 That the Laplace transform of this thing, and this the crux 22 00:01:13,349 --> 00:01:18,769 of the theorem, the Laplace transform of the convolution 23 00:01:18,769 --> 00:01:22,859 of these two functions is equal to the products of their 24 00:01:22,859 --> 00:01:29,030 Laplace transforms. It equals F of s, big capital F of s, 25 00:01:29,030 --> 00:01:32,969 times big capital G of s. 26 00:01:32,969 --> 00:01:36,049 Now, this might seem very abstract and very, you know, 27 00:01:36,049 --> 00:01:38,170 hard to kind of handle for you right now. 28 00:01:38,170 --> 00:01:39,829 So let's do an actual example. 29 00:01:39,829 --> 00:01:42,260 And actually, even better, let's do an inverse Laplace 30 00:01:42,260 --> 00:01:44,870 transform with an example. 31 00:01:44,870 --> 00:01:46,040 And actually, let me write one more thing. 32 00:01:46,040 --> 00:01:47,950 If this is true, then we could also do it the other way. 33 00:01:47,950 --> 00:01:51,579 We could also say that f-- and I'll just do it all in yellow; 34 00:01:51,579 --> 00:01:54,200 it takes me too much time to keep switching colors-- that 35 00:01:54,200 --> 00:01:58,799 the convolution of f and g, this is just a function of t, 36 00:01:58,799 --> 00:02:01,789 I can just say it's the inverse Laplace transform. 37 00:02:01,790 --> 00:02:06,480 It's just the inverse Laplace transform of F of 38 00:02:06,480 --> 00:02:07,540 s times G of s. 39 00:02:07,540 --> 00:02:08,680 Although I couldn't resist it. 40 00:02:08,680 --> 00:02:09,930 Let me switch colors. 41 00:02:09,930 --> 00:02:15,020 42 00:02:15,020 --> 00:02:16,420 There you go. 43 00:02:16,419 --> 00:02:18,569 Now, what good does all of this do? 44 00:02:18,569 --> 00:02:21,509 Well, we can take inverse Laplace transforms. Let's just 45 00:02:21,509 --> 00:02:25,759 say that I had-- let me write it down here-- let's say I 46 00:02:25,759 --> 00:02:29,079 told you that the following expression or function, let's 47 00:02:29,080 --> 00:02:34,430 say H of s-- let me write it this way-- H of s is equal to 48 00:02:34,430 --> 00:02:38,510 2s over s squared plus 1. 49 00:02:38,509 --> 00:02:40,879 Now, we did this long differential equations at the 50 00:02:40,879 --> 00:02:42,879 end, we end up with this thing and we have to take the 51 00:02:42,879 --> 00:02:45,169 inverse Laplace transform of it. 52 00:02:45,169 --> 00:02:50,579 So we want to figure out the inverse Laplace transform of H 53 00:02:50,580 --> 00:02:53,960 of s, or the inverse Laplace transform of 54 00:02:53,960 --> 00:02:55,830 this thing right there. 55 00:02:55,830 --> 00:03:02,090 So we want to figure out the inverse Laplace transform of 56 00:03:02,090 --> 00:03:09,284 this expression right here, 2s over s squared plus 1 squared. 57 00:03:09,284 --> 00:03:11,039 I don't want to lose that. 58 00:03:11,039 --> 00:03:12,169 Right there. 59 00:03:12,169 --> 00:03:15,759 Now, can we write this as the product of two Laplace 60 00:03:15,759 --> 00:03:18,000 transforms that we do know? 61 00:03:18,000 --> 00:03:19,509 Let's try to do it. 62 00:03:19,509 --> 00:03:21,789 So we can rewrite this. 63 00:03:21,789 --> 00:03:23,965 And so this is the inverse Laplace transform. 64 00:03:23,965 --> 00:03:26,879 So let me rewrite this expression down here. 65 00:03:26,879 --> 00:03:34,919 So I can rewrite 2s over s squared plus 1 squared. 66 00:03:34,919 --> 00:03:40,179 This is the same thing as-- let me write it this way-- 2 67 00:03:40,180 --> 00:03:51,420 times 1 over s squared plus 1, times s over s squared plus 1. 68 00:03:51,419 --> 00:03:52,489 I just kind of broke it up. 69 00:03:52,490 --> 00:03:54,590 If you multiply the numerators here, you get 2 times 70 00:03:54,590 --> 00:03:56,580 1, times s, or 2s. 71 00:03:56,580 --> 00:03:59,200 If you multiply the denominators here, s squared 72 00:03:59,199 --> 00:04:02,919 plus 1, times s squared plus 1, well, that's just s squared 73 00:04:02,919 --> 00:04:04,129 plus 1 squared. 74 00:04:04,129 --> 00:04:05,359 So this is the same thing. 75 00:04:05,360 --> 00:04:07,640 So if we want to take the inverse Laplace transform of 76 00:04:07,639 --> 00:04:11,529 this, it's the same thing as taking the inverse Laplace 77 00:04:11,530 --> 00:04:14,650 transform of this right here. 78 00:04:14,650 --> 00:04:17,649 Now, something should hopefully start 79 00:04:17,649 --> 00:04:18,870 popping out at you. 80 00:04:18,870 --> 00:04:22,360 If these were separate transforms, if they were on 81 00:04:22,360 --> 00:04:24,470 their own, we know what this is. 82 00:04:24,470 --> 00:04:28,380 If we call this F of s, if we said this is the Laplace 83 00:04:28,379 --> 00:04:31,230 transform of some function, we know what that function is. 84 00:04:31,230 --> 00:04:32,980 This is this piece right here. 85 00:04:32,980 --> 00:04:35,020 I'm just doing a little dotted line around it. 86 00:04:35,019 --> 00:04:37,810 This is the Laplace transform of sine of t. 87 00:04:37,810 --> 00:04:41,889 And then if we draw a little box around this one right 88 00:04:41,889 --> 00:04:44,560 here, this is the Laplace transform of 89 00:04:44,560 --> 00:04:47,480 cosine of t, G of s. 90 00:04:47,480 --> 00:04:51,110 So this is the Laplace transform of sine of t, or we 91 00:04:51,110 --> 00:04:55,509 could write that this implies that f of t is 92 00:04:55,509 --> 00:04:58,170 equal to sine of t. 93 00:04:58,170 --> 00:05:00,600 You should recognize that one by now. 94 00:05:00,600 --> 00:05:04,650 And this implies that g of t, if we define this as the 95 00:05:04,649 --> 00:05:10,649 Laplace transform of g, this means that g of t is equal to 96 00:05:10,649 --> 00:05:13,769 cosine of t. 97 00:05:13,769 --> 00:05:15,169 And, of course, when you take the inverse Laplace 98 00:05:15,170 --> 00:05:19,150 transforms, you could take the 2's out. 99 00:05:19,149 --> 00:05:21,439 So now what can we say? 100 00:05:21,439 --> 00:05:26,170 We can now say that the-- let me write it this way-- the 101 00:05:26,170 --> 00:05:28,110 inverse-- so actually, let me write it this way. 102 00:05:28,110 --> 00:05:30,889 Or, actually, a better thing to do, instead of taking the 2 103 00:05:30,889 --> 00:05:33,990 out, so I can leave it nice and clean, we could, if we 104 00:05:33,990 --> 00:05:40,220 were to draw a box around this whole thing, and define this 105 00:05:40,220 --> 00:05:46,470 whole thing as F of s, then F of s is the Laplace transform 106 00:05:46,470 --> 00:05:48,130 of 2 sine of t. 107 00:05:48,129 --> 00:05:49,500 I just wanted to include that 2. 108 00:05:49,500 --> 00:05:51,620 I didn't want to leave that out and confuse the issue. 109 00:05:51,620 --> 00:05:55,149 I wanted a very pure F of s times G of s. 110 00:05:55,149 --> 00:05:59,429 So this expression right here is the product of the Laplace 111 00:05:59,430 --> 00:06:04,199 transform of 2 sine of t, and the Laplace transform of 112 00:06:04,199 --> 00:06:05,610 cosine of t. 113 00:06:05,610 --> 00:06:11,400 Now, our convolution theorem told us this right here. 114 00:06:11,399 --> 00:06:14,509 That if we want to take the inverse Laplace transform of 115 00:06:14,509 --> 00:06:17,469 the Laplace transforms of two functions-- I know that sounds 116 00:06:17,470 --> 00:06:19,530 very confusing --but you just kind of pattern match. 117 00:06:19,529 --> 00:06:21,919 You say, OK look, this thing that I had here, I could 118 00:06:21,920 --> 00:06:24,689 rewrite it as a product of two Laplace 119 00:06:24,689 --> 00:06:26,459 transforms I can recognize. 120 00:06:26,459 --> 00:06:30,669 This right here is the Laplace transform of 2 sine of t. 121 00:06:30,670 --> 00:06:33,050 This is the Laplace transform of cosine of t. 122 00:06:33,050 --> 00:06:36,180 And we just wrote that as G of s, and F of s. 123 00:06:36,180 --> 00:06:40,040 So if I have an expression written like this, I can take 124 00:06:40,040 --> 00:06:42,439 the inverse Laplace transform and it'll be equal to the 125 00:06:42,439 --> 00:06:48,219 convolution of the original functions. 126 00:06:48,220 --> 00:06:51,830 It'll be equal to the convolution of the inverse of 127 00:06:51,829 --> 00:06:53,629 g or the inverse of f. 128 00:06:53,629 --> 00:06:54,750 Let me write it this way. 129 00:06:54,750 --> 00:06:56,810 I could write it like this. 130 00:06:56,810 --> 00:07:03,410 We know that f of t is equal to the inverse Laplace 131 00:07:03,410 --> 00:07:06,080 transform of F of s. 132 00:07:06,079 --> 00:07:09,829 And we know that g-- I should have done it in a different 133 00:07:09,829 --> 00:07:14,519 color, but I'll do g in green-- we know that g of t is 134 00:07:14,519 --> 00:07:21,189 equal to the inverse Laplace transform of G of s. 135 00:07:21,189 --> 00:07:24,449 So we can rewrite the convolution theorem as the 136 00:07:24,449 --> 00:07:27,959 inverse-- and this might maybe confuse you more than help, 137 00:07:27,959 --> 00:07:30,060 but I'll give my best shot. 138 00:07:30,060 --> 00:07:34,550 The inverse Laplace transform of-- and I'll try to stay true 139 00:07:34,550 --> 00:07:43,480 to the colors-- of F of s times G of s is equal to-- I'm 140 00:07:43,480 --> 00:07:45,220 just restating this convolution 141 00:07:45,220 --> 00:07:45,690 theorem right here. 142 00:07:45,689 --> 00:07:49,910 This is equal to the convolution of the inverse 143 00:07:49,910 --> 00:07:52,340 Laplace transform of F of s. 144 00:07:52,339 --> 00:07:55,679 So it's equal to the convolution of the inverse 145 00:07:55,680 --> 00:08:04,620 Laplace transform of F of s with the inverse Laplace 146 00:08:04,620 --> 00:08:06,480 transform of G of s. 147 00:08:06,480 --> 00:08:10,379 With the inverse Laplace transform of 148 00:08:10,379 --> 00:08:15,409 capital G, of G of s. 149 00:08:15,410 --> 00:08:17,740 I'm not sure if that helps you or not, but if you go back to 150 00:08:17,740 --> 00:08:19,160 this example it might. 151 00:08:19,160 --> 00:08:22,260 This is F of s, this is F of s right here. 152 00:08:22,259 --> 00:08:28,129 2 times-- I'll do it in the light blue-- this is 2 over s 153 00:08:28,129 --> 00:08:29,319 squared plus 1. 154 00:08:29,319 --> 00:08:32,460 That's F of s in our example. 155 00:08:32,460 --> 00:08:39,820 And the G of s was s over s squared plus 1. 156 00:08:39,820 --> 00:08:42,590 And all I got that from is I just broke this up into two 157 00:08:42,590 --> 00:08:43,570 things that I recognize. 158 00:08:43,570 --> 00:08:46,980 If I multiply this together, I get back to my original thing 159 00:08:46,980 --> 00:08:49,539 that I was trying to take the inverse Laplace transform of. 160 00:08:49,539 --> 00:08:52,929 And so the convolution theorem just says that, OK, well, the 161 00:08:52,929 --> 00:08:57,169 inverse Laplace transform of this is equal to the inverse 162 00:08:57,169 --> 00:09:05,089 Laplace transform of 2 over s squared plus 1, convoluted 163 00:09:05,090 --> 00:09:14,960 with the inverse Laplace transform of our G of s, of s 164 00:09:14,960 --> 00:09:18,430 over s squared plus 1. 165 00:09:18,429 --> 00:09:20,429 And we know what these things are. 166 00:09:20,429 --> 00:09:22,899 I already told them to you, but they should be somewhat 167 00:09:22,899 --> 00:09:23,699 second nature now. 168 00:09:23,700 --> 00:09:25,650 This is 2 times sine of t. 169 00:09:25,649 --> 00:09:28,720 170 00:09:28,720 --> 00:09:31,110 You take the Laplace transform of sine of t, you get 1 over s 171 00:09:31,110 --> 00:09:33,190 squared plus 1, and then you multiply it by 2, you get the 172 00:09:33,190 --> 00:09:34,260 2 up there. 173 00:09:34,259 --> 00:09:37,870 And you're going to have to convolute that with the 174 00:09:37,870 --> 00:09:39,659 inverse Laplace transform of this thing here. 175 00:09:39,659 --> 00:09:40,579 And we already went over this. 176 00:09:40,580 --> 00:09:41,900 This is cosine of t. 177 00:09:41,899 --> 00:09:44,669 178 00:09:44,669 --> 00:09:49,479 So our result so far-- let me be very clear. 179 00:09:49,480 --> 00:09:52,129 It's always good to take a step back and just think about 180 00:09:52,129 --> 00:09:55,269 what we're doing, much less why we're doing it. 181 00:09:55,269 --> 00:10:00,960 But let's see, the inverse Laplace transform of this 182 00:10:00,960 --> 00:10:06,650 thing up in this top left corner, 2s over s squared plus 183 00:10:06,649 --> 00:10:11,220 1 squared, which before we did what we're doing now was very 184 00:10:11,220 --> 00:10:13,740 hard to figure out-- actually, this would be a curly bracket 185 00:10:13,740 --> 00:10:17,620 right here, but you get the idea-- is equal to this. 186 00:10:17,620 --> 00:10:30,710 It's equal to 2 sine of t, convoluted with cosine of t. 187 00:10:30,710 --> 00:10:32,830 And you're like, Sal, throughout this whole process 188 00:10:32,830 --> 00:10:34,990 I've already forgotten what it means to convolute two 189 00:10:34,990 --> 00:10:37,330 functions, so let's convolute them. 190 00:10:37,330 --> 00:10:39,570 And I'll just write the definition, or the definition 191 00:10:39,570 --> 00:10:41,310 we're using of the convolution. 192 00:10:41,309 --> 00:10:44,369 That f convoluted with g-- it's going to be 193 00:10:44,370 --> 00:10:45,039 a function of g. 194 00:10:45,039 --> 00:10:46,349 I'll just write this short-hand-- is equal to the 195 00:10:46,350 --> 00:10:51,570 integral from 0 to t, of f of t minus tau, 196 00:10:51,570 --> 00:10:54,910 times g of tau, dtau. 197 00:10:54,909 --> 00:11:03,279 So 2 sine of t convoluted with cosine of t is equal to-- let 198 00:11:03,279 --> 00:11:09,574 me do a neutral color-- the integral from 0 to t, of 2 199 00:11:09,575 --> 00:11:22,290 sine of t, minus tau, times the cosine of tau, dtau. 200 00:11:22,289 --> 00:11:25,799 Now if you watched the very last video I made, I actually 201 00:11:25,799 --> 00:11:29,514 solved this, or I solved a very similar thing to this. 202 00:11:29,514 --> 00:11:34,399 If we take the 2 out we get 2, times the integral from 0 to 203 00:11:34,399 --> 00:11:43,090 t, of sine of t minus tau, times the cosine of tau. 204 00:11:43,090 --> 00:11:45,950 I actually solved this in the previous video. 205 00:11:45,950 --> 00:11:48,490 This right here, this is the convolution of sine of t and 206 00:11:48,490 --> 00:11:49,169 cosine of t. 207 00:11:49,169 --> 00:11:54,000 It's sine of t convoluted with cosine of t. 208 00:11:54,000 --> 00:11:56,870 And I show you in the previous video, just watch that video, 209 00:11:56,870 --> 00:11:59,490 where I introduce a convolution, that this thing 210 00:11:59,490 --> 00:12:05,870 right here is equal to 1/2t sine of t. 211 00:12:05,870 --> 00:12:08,460 Now, if this thing is equal to 1/2 t sine of t, and I have to 212 00:12:08,460 --> 00:12:14,750 multiply it by 2, then we get, our big result, that the 213 00:12:14,750 --> 00:12:21,549 inverse Laplace transform of 2s over s squared plus 1 214 00:12:21,549 --> 00:12:26,169 squared is equal to the convolution of 2 sine of t 215 00:12:26,169 --> 00:12:27,479 with cosine of t. 216 00:12:27,480 --> 00:12:31,550 Which is just 2 times this thing here, which is 2 times 217 00:12:31,549 --> 00:12:37,649 1/2-- those cancel out-- so it equals t sine of t. 218 00:12:37,649 --> 00:12:39,470 And once you get the hang of it, you won't have to go 219 00:12:39,470 --> 00:12:40,820 through all of these steps. 220 00:12:40,820 --> 00:12:43,760 But the key is to recognize that this could be broken down 221 00:12:43,759 --> 00:12:47,799 as the products of two Laplace transforms that you recognize. 222 00:12:47,799 --> 00:12:50,159 This could be broken down as the product of two Laplace 223 00:12:50,159 --> 00:12:51,839 transforms we recognized. 224 00:12:51,840 --> 00:12:53,629 This is the Laplace transform of 2 sine of t. 225 00:12:53,629 --> 00:12:56,009 This was the Laplace transform of cosine of t. 226 00:12:56,009 --> 00:12:59,319 So the inverse Laplace transform of our original 227 00:12:59,320 --> 00:13:02,460 thing, or original expression, is just the convolution of 228 00:13:02,460 --> 00:13:04,759 that with that. 229 00:13:04,759 --> 00:13:07,189 And if you watched the previous video, you'd realize 230 00:13:07,190 --> 00:13:10,160 that actually calculating that convolution was no simple 231 00:13:10,159 --> 00:13:11,889 task, but it can be done. 232 00:13:11,889 --> 00:13:13,840 So you actually can get an integral form. 233 00:13:13,840 --> 00:13:16,200 Even if it can't be done, you can get your answer, at least, 234 00:13:16,200 --> 00:13:19,509 in terms of some integral. 235 00:13:19,509 --> 00:13:22,919 So I haven't proven the convolution 236 00:13:22,919 --> 00:13:23,729 theorem to you just yet. 237 00:13:23,730 --> 00:13:25,259 I'll do that in a future video. 238 00:13:25,259 --> 00:13:30,539 But hopefully, this gave you a little bit of a sense of how 239 00:13:30,539 --> 00:13:33,039 you can use it to actually take inverse Laplace 240 00:13:33,039 --> 00:13:34,839 transforms. And remember, the reason why we're learning to 241 00:13:34,840 --> 00:13:37,040 take inverse Laplace transforms, and we have all of 242 00:13:37,039 --> 00:13:40,279 these tools to do it, is because that's always that 243 00:13:40,279 --> 00:13:42,949 last step when you're solving these differential equations, 244 00:13:42,950 --> 00:13:45,490 using your Laplace transforms. 245 00:13:45,490 --> 00:13:45,647