1 00:00:00,000 --> 00:00:00,680 2 00:00:00,680 --> 00:00:03,540 Well, now's as good a time as any to go over some 3 00:00:03,540 --> 00:00:06,669 interesting and very useful properties 4 00:00:06,669 --> 00:00:07,669 of the Laplace transform. 5 00:00:07,669 --> 00:00:10,620 And the first is to show that it is a linear operator. 6 00:00:10,619 --> 00:00:11,229 And what does that mean? 7 00:00:11,230 --> 00:00:17,839 Well, let's say I wanted to take the Laplace transform of 8 00:00:17,839 --> 00:00:19,269 the sum of the-- we call it the 9 00:00:19,269 --> 00:00:20,589 weighted sum of two functions. 10 00:00:20,589 --> 00:00:25,222 So say some constant, c1, times my first function, f of 11 00:00:25,222 --> 00:00:30,600 t, plus some constant, c2, times my second 12 00:00:30,600 --> 00:00:31,850 function, g of t. 13 00:00:31,850 --> 00:00:34,560 14 00:00:34,560 --> 00:00:38,590 Well, by the definition of the Laplace transform, this would 15 00:00:38,590 --> 00:00:45,280 be equal to the improper integral from 0 to infinity of 16 00:00:45,280 --> 00:00:51,810 e to the minus st, times whatever our function that 17 00:00:51,810 --> 00:00:56,219 we're taking the Laplace transform of, so times c1, f 18 00:00:56,219 --> 00:01:01,479 of t, plus c2, g of t-- I think you know where this is 19 00:01:01,479 --> 00:01:04,409 going-- all of that dt. 20 00:01:04,409 --> 00:01:09,819 And then that is equal to the integral from 0 to infinity. 21 00:01:09,819 --> 00:01:12,779 Let's just distribute the e the minus st. 22 00:01:12,780 --> 00:01:15,349 That is equal to what? 23 00:01:15,349 --> 00:01:29,954 That is equal to c1e to the minus st, f of t, plus c2e to 24 00:01:29,954 --> 00:01:38,560 the minus st, g of t, and all of that times dt. 25 00:01:38,560 --> 00:01:42,219 And just by the definition of how the properties of 26 00:01:42,219 --> 00:01:44,519 integrals work, we know that we can split this up into two 27 00:01:44,519 --> 00:01:45,649 integrals, right? 28 00:01:45,650 --> 00:01:49,170 If the integral of the sum of two functions is equal to the 29 00:01:49,170 --> 00:01:50,579 sum of their integrals. 30 00:01:50,579 --> 00:01:51,859 And these are just constant. 31 00:01:51,859 --> 00:01:57,189 So this is going to be equal to c1 times the integral from 32 00:01:57,189 --> 00:02:04,859 0 to infinity of e to the minus st, times f of t, d of 33 00:02:04,859 --> 00:02:12,960 t, plus c2 times the integral from 0 to infinity of e to the 34 00:02:12,960 --> 00:02:17,840 minus st, g of t, dt. 35 00:02:17,840 --> 00:02:21,879 And this was just a very long-winded way of saying, 36 00:02:21,879 --> 00:02:22,409 what is this? 37 00:02:22,409 --> 00:02:24,409 This is the Laplace transform of f of t. 38 00:02:24,409 --> 00:02:26,060 This is the Laplace transform of g of t. 39 00:02:26,060 --> 00:02:32,550 So this is equal to c1 times the Laplace transform of f of 40 00:02:32,550 --> 00:02:40,430 t, plus c2 times-- this is the Laplace transform-- the 41 00:02:40,430 --> 00:02:45,110 Laplace transform of g of t. 42 00:02:45,110 --> 00:02:48,730 And so, we have just shown that the Laplace transform is 43 00:02:48,729 --> 00:02:50,209 a linear operator, right? 44 00:02:50,210 --> 00:02:53,030 The Laplace transform of this is equal to this. 45 00:02:53,030 --> 00:02:57,310 So essentially, you can kind of break up the sum and take 46 00:02:57,310 --> 00:02:59,830 out the constants, and just take the Laplace transform. 47 00:02:59,830 --> 00:03:01,719 That's something useful to know, and you might have 48 00:03:01,719 --> 00:03:03,259 guessed that was the case anyway. 49 00:03:03,259 --> 00:03:05,149 But now you know for sure. 50 00:03:05,150 --> 00:03:08,640 Now we'll do something which I consider even more 51 00:03:08,639 --> 00:03:09,419 interesting. 52 00:03:09,419 --> 00:03:12,729 And this is actually going to be a big clue as to why 53 00:03:12,729 --> 00:03:17,359 Laplace transforms are extremely useful for solving 54 00:03:17,360 --> 00:03:19,020 differential equations. 55 00:03:19,020 --> 00:03:26,120 So let's say I want to find the Laplace transform of f 56 00:03:26,120 --> 00:03:28,379 prime of t. 57 00:03:28,379 --> 00:03:31,049 So I have some f of t, I take its derivative, and then I 58 00:03:31,050 --> 00:03:33,100 want the Laplace transform of that. 59 00:03:33,099 --> 00:03:35,439 Let's see if we can find a relationship between the 60 00:03:35,439 --> 00:03:37,840 Laplace transform of the derivative of a function, and 61 00:03:37,840 --> 00:03:40,950 the Laplace transform of the function. 62 00:03:40,949 --> 00:03:44,179 So we're going to use some integration by parts here. 63 00:03:44,180 --> 00:03:46,700 64 00:03:46,699 --> 00:03:48,000 Let me just say what this is, first of all. 65 00:03:48,000 --> 00:03:55,360 This is equal to the integral from 0 to infinity of e to the 66 00:03:55,360 --> 00:04:03,390 minus st, times f prime of t, dt. 67 00:04:03,389 --> 00:04:05,939 And to solve this, we're going to use integration by parts. 68 00:04:05,939 --> 00:04:07,491 Let me write it in the corner, just so you 69 00:04:07,491 --> 00:04:08,179 remember what it is. 70 00:04:08,180 --> 00:04:11,200 So I think I memorized it, because I recorded that last 71 00:04:11,199 --> 00:04:13,304 video not too long ago. 72 00:04:13,305 --> 00:04:14,200 I'm just going to write this shorthand. 73 00:04:14,199 --> 00:04:20,149 The integral of u-- well, let's say uv prime, because 74 00:04:20,149 --> 00:04:25,389 that will match what we have up here better-- is equal to 75 00:04:25,389 --> 00:04:29,789 both functions without the derivitives, uv minus the 76 00:04:29,790 --> 00:04:30,939 integral of the opposite. 77 00:04:30,939 --> 00:04:32,719 So the opposite is u prime v. 78 00:04:32,720 --> 00:04:35,550 79 00:04:35,550 --> 00:04:38,550 So here, the substitution is pretty clear, right? 80 00:04:38,550 --> 00:04:43,460 Because we want to end up with f of x, right? 81 00:04:43,459 --> 00:04:46,879 So let's make v prime is f prime, and let's make u e to 82 00:04:46,879 --> 00:04:49,170 the minus st. So let's do that. 83 00:04:49,170 --> 00:04:58,140 u is going to be e to the minus st, and v is going to 84 00:04:58,139 --> 00:04:59,909 equal what? 85 00:04:59,910 --> 00:05:05,150 v is going to equal f prime of t. 86 00:05:05,149 --> 00:05:14,069 And then u prime would be minus se to the minus st. And 87 00:05:14,069 --> 00:05:21,699 then, v prime-- oh, sorry, this is v prime, right? 88 00:05:21,699 --> 00:05:26,620 v prime is f prime of t, so v is just going to be 89 00:05:26,620 --> 00:05:27,720 equal to f of t. 90 00:05:27,720 --> 00:05:30,090 I hope I didn't say that wrong the first time. 91 00:05:30,089 --> 00:05:31,269 But you see what I'm saying. 92 00:05:31,269 --> 00:05:37,649 This is u, that's u, and this is v prime. 93 00:05:37,649 --> 00:05:39,319 And if this is v prime, then if you were to take the 94 00:05:39,319 --> 00:05:42,069 antiderivative of both sides, then v is equal to f of t. 95 00:05:42,069 --> 00:05:45,490 So let's apply integration by parts. 96 00:05:45,490 --> 00:05:51,030 So this Laplace transform, which is this, is equal to uv, 97 00:05:51,029 --> 00:06:01,469 which is equal to e to the minus st, times v, f of t, 98 00:06:01,470 --> 00:06:08,150 minus the integral-- and, of course, we're going to have to 99 00:06:08,149 --> 00:06:10,569 evaluate this from 0 to infinity. 100 00:06:10,569 --> 00:06:15,219 I'll keep the improper integral with 101 00:06:15,220 --> 00:06:15,960 us the whole time. 102 00:06:15,959 --> 00:06:19,379 I won't switch back and forth between the definite and 103 00:06:19,379 --> 00:06:20,310 indefinite integral. 104 00:06:20,310 --> 00:06:22,589 So minus this part. 105 00:06:22,589 --> 00:06:28,649 So the integral from 0 to infinity of u prime. 106 00:06:28,649 --> 00:06:36,639 u prime is minus se to the minus st times v-- 107 00:06:36,639 --> 00:06:43,589 v is f of t-- dt. 108 00:06:43,589 --> 00:06:43,789 Now, let's see. 109 00:06:43,790 --> 00:06:45,275 We have a minus and a minus, let's make 110 00:06:45,274 --> 00:06:47,389 both of these pluses. 111 00:06:47,389 --> 00:06:49,579 This s is just a constant, so we can bring it out. 112 00:06:49,579 --> 00:07:02,149 So that is equal to e to the minus st, f of t, evaluated 113 00:07:02,149 --> 00:07:07,549 from 0 to infinity, or as we approach infinity, plus s 114 00:07:07,550 --> 00:07:13,889 times the integral from 0 to infinity of e to the minus st, 115 00:07:13,889 --> 00:07:17,110 f of t, dt. 116 00:07:17,110 --> 00:07:19,069 And here, we see, what is this? 117 00:07:19,069 --> 00:07:22,230 This is the Laplace transform of f of t, right? 118 00:07:22,230 --> 00:07:25,430 119 00:07:25,430 --> 00:07:27,009 Let's evaluate this part. 120 00:07:27,009 --> 00:07:29,310 So when we evaluated in infinity, as we approach 121 00:07:29,310 --> 00:07:34,350 infinity, e to the minus infinity approaches 0. 122 00:07:34,350 --> 00:07:38,960 f of infinity-- now this is an interesting question. 123 00:07:38,959 --> 00:07:40,180 f of infinity-- I don't know. 124 00:07:40,180 --> 00:07:43,569 That could be large, that could be small, that 125 00:07:43,569 --> 00:07:46,399 approaches some value, right? 126 00:07:46,399 --> 00:07:48,000 This approach 0, so we're not sure. 127 00:07:48,000 --> 00:07:51,970 If this increases faster than this approaches 0, then this 128 00:07:51,970 --> 00:07:53,240 will diverge. 129 00:07:53,240 --> 00:07:55,920 I won't go into the mathematics of whether this 130 00:07:55,920 --> 00:07:59,180 converges or diverges, but let's just say, in very rough 131 00:07:59,180 --> 00:08:05,030 terms, that this will converge to 0 if f of t grows slower 132 00:08:05,029 --> 00:08:08,000 than e to the minus st shrinks. 133 00:08:08,000 --> 00:08:11,939 And maybe later on we'll do some more rigorous definitions 134 00:08:11,939 --> 00:08:16,300 of under what conditions will this 135 00:08:16,300 --> 00:08:17,810 expression actually converge. 136 00:08:17,810 --> 00:08:23,290 But let's assume that f of t grows slower than e to the st, 137 00:08:23,290 --> 00:08:31,310 or it diverges slower than this converges, is another way 138 00:08:31,310 --> 00:08:31,910 to view it. 139 00:08:31,910 --> 00:08:35,070 Or this grows slower than this shrinks. 140 00:08:35,070 --> 00:08:37,730 So if this grows slower than this shrinks, then this whole 141 00:08:37,730 --> 00:08:41,320 expression will approach 0. 142 00:08:41,320 --> 00:08:43,730 And then you want to subtract this whole expression 143 00:08:43,730 --> 00:08:45,180 evaluated at 0. 144 00:08:45,179 --> 00:08:52,804 So e to the 0 is 1 times f of 0-- so that's just f of 0-- 145 00:08:52,804 --> 00:08:57,029 plus s times-- we said, this is the Laplace transform of f 146 00:08:57,029 --> 00:08:58,809 of t, that's our definition-- so the Laplace 147 00:08:58,809 --> 00:09:05,579 transform of f of t. 148 00:09:05,580 --> 00:09:07,120 And now we have an interesting property. 149 00:09:07,120 --> 00:09:09,029 What was the left-hand side of everything we were doing? 150 00:09:09,029 --> 00:09:11,740 The Laplace transform of f prime of t. 151 00:09:11,740 --> 00:09:14,860 So let me just write all over again. 152 00:09:14,860 --> 00:09:15,850 And I'll switch colors. 153 00:09:15,850 --> 00:09:25,710 The Laplace transform of f prime of t is equal to s times 154 00:09:25,710 --> 00:09:36,740 the Laplace transform of f of t minus f of 0. 155 00:09:36,740 --> 00:09:39,480 And now, let's just extend this further. 156 00:09:39,480 --> 00:09:41,090 What is the Laplace transform-- and this is a 157 00:09:41,090 --> 00:09:44,800 really useful thing to know-- what is the Laplace transform 158 00:09:44,799 --> 00:09:49,029 of f prime prime of t? 159 00:09:49,029 --> 00:09:51,269 Well, we can do a little pattern matching here, right? 160 00:09:51,269 --> 00:09:54,960 That's going to be s times the Laplace transform of its 161 00:09:54,960 --> 00:09:59,000 antiderivative, times the Laplace transform of f prime 162 00:09:59,000 --> 00:10:02,750 of t, right? 163 00:10:02,750 --> 00:10:04,019 This goes to this, that's an antiderivative. 164 00:10:04,019 --> 00:10:07,799 This goes to this, that's one antiderivative. 165 00:10:07,799 --> 00:10:13,519 Minus f prime of 0, right? 166 00:10:13,519 --> 00:10:16,779 But then what's the Laplace transform of this? 167 00:10:16,779 --> 00:10:21,720 This is going to be equal to s times the Laplace transform of 168 00:10:21,720 --> 00:10:23,660 f prime of t, but what's that? 169 00:10:23,659 --> 00:10:26,559 That's this, right? 170 00:10:26,559 --> 00:10:35,549 That's s times the Laplace transform of f of t, minus f 171 00:10:35,549 --> 00:10:38,269 of 0, right? 172 00:10:38,269 --> 00:10:40,360 I just substituted this with this. 173 00:10:40,360 --> 00:10:44,750 Minus f prime of 0. 174 00:10:44,750 --> 00:10:47,379 And we get the Laplace transform of the second 175 00:10:47,379 --> 00:10:51,700 derivative is equal to s squared times the Laplace 176 00:10:51,700 --> 00:11:01,590 transform of our function, f of t, minus s times f of 0, 177 00:11:01,590 --> 00:11:04,430 minus f prime of 0. 178 00:11:04,429 --> 00:11:05,839 And I think you're starting to see a pattern here. 179 00:11:05,840 --> 00:11:10,530 This is the Laplace transform of f prime prime of t. 180 00:11:10,529 --> 00:11:12,240 And I think you're starting to see why the Laplace 181 00:11:12,240 --> 00:11:13,389 transform is useful. 182 00:11:13,389 --> 00:11:17,909 It turns derivatives into multiplications by f. 183 00:11:17,909 --> 00:11:20,509 And actually, as you'll see later, it turns integration to 184 00:11:20,509 --> 00:11:22,330 divisions by s. 185 00:11:22,330 --> 00:11:26,070 And you can take arbitrary derivatives and just keep 186 00:11:26,070 --> 00:11:27,290 multiplying by s. 187 00:11:27,289 --> 00:11:28,000 And you see this pattern. 188 00:11:28,000 --> 00:11:29,610 And I'm running out of time. 189 00:11:29,610 --> 00:11:31,310 But I'll leave it up to you to figure out what the Laplace 190 00:11:31,309 --> 00:11:33,789 transform of the third derivative of f is. 191 00:11:33,789 --> 00:11:35,990 See you in the next video. 192 00:11:35,990 --> 00:11:36,000