1 00:00:00,000 --> 00:00:00,730 2 00:00:00,730 --> 00:00:03,540 The whole point in learning differential equations is that 3 00:00:03,540 --> 00:00:06,269 eventually we want to model real physical systems. I know 4 00:00:06,269 --> 00:00:09,330 everything we've done so far has really just been a toolkit 5 00:00:09,330 --> 00:00:12,390 of being able to solve them, but the whole reason is that 6 00:00:12,390 --> 00:00:15,410 because differential equations can describe a lot of systems, 7 00:00:15,410 --> 00:00:17,320 and then we can actually model them that way. 8 00:00:17,320 --> 00:00:20,750 And we know that in the real world, everything isn't these 9 00:00:20,750 --> 00:00:23,820 nice continuous functions, so over the next couple of videos 10 00:00:23,820 --> 00:00:25,890 we're going to talk about functions that are a little 11 00:00:25,890 --> 00:00:28,880 bit more discontinuous than what you might be used to even 12 00:00:28,879 --> 00:00:31,460 in kind of a traditional calculus or traditional 13 00:00:31,460 --> 00:00:32,880 Precalculus class. 14 00:00:32,880 --> 00:00:36,910 And the first one is the unit step function. 15 00:00:36,909 --> 00:00:39,789 Let's write it as u, and then I'll put a little 16 00:00:39,789 --> 00:00:43,289 subscript c here of t. 17 00:00:43,289 --> 00:00:48,109 And it's defined as when t is-- let me put it this way. 18 00:00:48,109 --> 00:00:52,149 It's defined as 0 when t is less than-- whatever subscript 19 00:00:52,149 --> 00:00:55,219 I put here-- when t is less than c. 20 00:00:55,219 --> 00:00:58,379 And it's defined as 1-- that's why we call it the unit step 21 00:00:58,380 --> 00:01:02,900 function-- when t is greater than or equal to c. 22 00:01:02,899 --> 00:01:05,700 And if I had to graph this, and you could graph it as well 23 00:01:05,700 --> 00:01:07,030 but it's not too difficult to graph. 24 00:01:07,030 --> 00:01:10,621 Let me draw my x-axis right here. 25 00:01:10,621 --> 00:01:13,010 And I'll do a little thicker line. 26 00:01:13,010 --> 00:01:15,490 That's my x-axis right there. 27 00:01:15,489 --> 00:01:20,429 This is my y-axis right there. 28 00:01:20,430 --> 00:01:22,990 29 00:01:22,989 --> 00:01:25,379 And when we talk about Laplace transforms, which we'll talk 30 00:01:25,379 --> 00:01:28,399 about shortly, we only care about t is greater than 0. 31 00:01:28,400 --> 00:01:30,090 Because we saw, in our definition of the Laplace 32 00:01:30,090 --> 00:01:32,510 transform, we're always taking the integral from 0 to 33 00:01:32,510 --> 00:01:34,359 infinity, so we're only dealing with 34 00:01:34,359 --> 00:01:36,469 the positive x-axis. 35 00:01:36,469 --> 00:01:39,159 But anyway, by this definition, it would be zero 36 00:01:39,159 --> 00:01:43,969 all the way until you get to some value c, so you'd be zero 37 00:01:43,969 --> 00:01:45,400 until you get to c. 38 00:01:45,400 --> 00:01:49,460 And then at c, you jump, and the point c is included x is 39 00:01:49,459 --> 00:01:51,219 equal to c here. 40 00:01:51,219 --> 00:01:53,069 So it's included, so I'll put a dot there, because it's 41 00:01:53,069 --> 00:01:54,299 greater than or equal to c. 42 00:01:54,299 --> 00:01:57,039 You're at 1, so this is 1 right here. 43 00:01:57,040 --> 00:01:59,790 And then you go forward for all of time. 44 00:01:59,790 --> 00:02:01,960 And you're like, Sal, you just said that differential 45 00:02:01,959 --> 00:02:05,099 equations, we're learning to model things, why is this type 46 00:02:05,099 --> 00:02:06,759 of a function useful? 47 00:02:06,760 --> 00:02:08,580 Well, in the real world, sometimes you do have 48 00:02:08,580 --> 00:02:11,100 something that essentially jolts something, that moves it 49 00:02:11,099 --> 00:02:12,789 from this position to that position. 50 00:02:12,789 --> 00:02:15,449 And obviously, nothing can move it immediately like this, 51 00:02:15,449 --> 00:02:18,649 but you might have some system, it could be an 52 00:02:18,650 --> 00:02:22,580 electrical system or mechanical system, where maybe 53 00:02:22,580 --> 00:02:25,000 the behavior looks something like this, where maybe it 54 00:02:25,000 --> 00:02:26,680 moves it like that or something. 55 00:02:26,680 --> 00:02:30,090 And this function is a pretty good analytic approximation 56 00:02:30,090 --> 00:02:32,490 for some type of real world behavior like this when 57 00:02:32,490 --> 00:02:33,920 something just gets moved. 58 00:02:33,919 --> 00:02:36,009 Whenever we solve these differential equations 59 00:02:36,009 --> 00:02:38,539 analytically, we're really just trying to get a pure 60 00:02:38,539 --> 00:02:39,729 model of something. 61 00:02:39,729 --> 00:02:42,609 Eventually, we'll see that it doesn't perfectly describe 62 00:02:42,610 --> 00:02:44,730 things, but it helps describe it enough for us to get a 63 00:02:44,729 --> 00:02:47,039 sense of what's going to happen. 64 00:02:47,039 --> 00:02:49,819 Sometimes it will completely describe things, but anyway, 65 00:02:49,819 --> 00:02:53,219 we can ignore that for now, so let me get rid of these things 66 00:02:53,219 --> 00:02:54,030 right there. 67 00:02:54,030 --> 00:02:56,180 So the first question is, well, you know, what if 68 00:02:56,180 --> 00:02:58,000 something doesn't jar just like that? 69 00:02:58,000 --> 00:03:03,500 What if I want to construct more fancy unit functions or 70 00:03:03,500 --> 00:03:05,270 more fancy step functions? 71 00:03:05,270 --> 00:03:07,469 Let's say I wanted to construct something that 72 00:03:07,469 --> 00:03:09,349 looked like this. 73 00:03:09,349 --> 00:03:13,939 Let me say this is my y-axis. 74 00:03:13,939 --> 00:03:16,310 This is my x-axis. 75 00:03:16,310 --> 00:03:18,699 And let's say I wanted to construct something that is 76 00:03:18,699 --> 00:03:23,819 at-- and let me do it in a different color. 77 00:03:23,819 --> 00:03:29,935 Let's say it's at 2 until I get to pi. 78 00:03:29,935 --> 00:03:33,449 79 00:03:33,449 --> 00:03:40,409 And then from pi until forever it just stays at zero. 80 00:03:40,409 --> 00:03:45,289 So how could I construct this function right here using my 81 00:03:45,289 --> 00:03:47,349 unit step function? 82 00:03:47,349 --> 00:03:51,939 So what if I had written it as-- so my unit step 83 00:03:51,939 --> 00:03:57,629 function's at zero initially, so what if I make it 2 minus a 84 00:03:57,629 --> 00:04:04,099 unit step function that starts at pi? 85 00:04:04,099 --> 00:04:06,769 86 00:04:06,770 --> 00:04:10,710 So if I define my function here, will this work? 87 00:04:10,710 --> 00:04:13,250 Well, this unit step function, when we pass pi, is only going 88 00:04:13,250 --> 00:04:14,969 to be equal to 1, but we want this thing 89 00:04:14,969 --> 00:04:15,900 to be equal to zero. 90 00:04:15,900 --> 00:04:18,819 So it has to be 2 minus 2, so I'll have to put at 2 here, 91 00:04:18,819 --> 00:04:20,009 and this should work. 92 00:04:20,009 --> 00:04:26,730 When we're at any value below pi, when t is less than pi 93 00:04:26,730 --> 00:04:29,970 here, this becomes a zero, so our function will just 94 00:04:29,970 --> 00:04:32,870 evaluate to 2, which is right there. 95 00:04:32,870 --> 00:04:36,490 But as soon as we hit t is equal to pi, that pi is the c 96 00:04:36,490 --> 00:04:40,050 in this example, as soon as we hit that, the unit step 97 00:04:40,050 --> 00:04:41,150 function becomes 1. 98 00:04:41,149 --> 00:04:45,120 We multiply that by 2, and we have 2 minus 2, and then we 99 00:04:45,120 --> 00:04:48,649 end up here with zero, 100 00:04:48,649 --> 00:04:50,810 Now, that might be nice and everything, but let's say you 101 00:04:50,810 --> 00:04:54,610 wanted for it to go back up. 102 00:04:54,610 --> 00:05:03,600 Let's say that instead of it going like this-- let me kind 103 00:05:03,600 --> 00:05:07,180 of erase that by overdrawing the x-axis again-- we want the 104 00:05:07,180 --> 00:05:09,509 function to jump up again. 105 00:05:09,509 --> 00:05:10,860 We want it to jump up again. 106 00:05:10,860 --> 00:05:14,150 And lets say at some value, let's say it's at 2pi, we want 107 00:05:14,149 --> 00:05:16,209 the function to jump up again. 108 00:05:16,209 --> 00:05:18,919 How could we construct this? 109 00:05:18,920 --> 00:05:20,629 And we could make it jump to anything, but let's say we 110 00:05:20,629 --> 00:05:22,409 want it to jump back to 2. 111 00:05:22,410 --> 00:05:25,290 Well, we could just add another unit step function 112 00:05:25,290 --> 00:05:28,810 here, something that would have been zero all along, all 113 00:05:28,810 --> 00:05:30,269 the way up until this point. 114 00:05:30,269 --> 00:05:34,159 But then at 2pi, it jumps, so in this case, 115 00:05:34,160 --> 00:05:36,475 our c would be 2pi. 116 00:05:36,475 --> 00:05:38,670 That's our unit step function, and we want it to jump to 2. 117 00:05:38,670 --> 00:05:40,430 This would just jump to 1 by itself. 118 00:05:40,430 --> 00:05:42,069 So let's multiply it by 2. 119 00:05:42,069 --> 00:05:43,480 And now we have this function. 120 00:05:43,480 --> 00:05:45,780 So you could imagine, you can make an arbitrarily 121 00:05:45,779 --> 00:05:49,179 complicated function of things jumping up and down to 122 00:05:49,180 --> 00:05:52,259 different levels based on different essentially linear 123 00:05:52,259 --> 00:05:54,719 combinations of these unit step functions. 124 00:05:54,720 --> 00:05:56,090 Now, what if I wanted to do something 125 00:05:56,089 --> 00:05:57,029 a little bit fancier? 126 00:05:57,029 --> 00:05:59,989 What if I wanted to do something that-- let's say I 127 00:05:59,990 --> 00:06:05,420 have some function that looks like this. 128 00:06:05,420 --> 00:06:07,756 Let me draw some function. 129 00:06:07,755 --> 00:06:09,339 I should draw straighter than that. 130 00:06:09,339 --> 00:06:11,949 I should have some standards. 131 00:06:11,949 --> 00:06:16,860 So let's say that just my regular f of t-- 132 00:06:16,860 --> 00:06:18,740 let me, this is x. 133 00:06:18,740 --> 00:06:20,030 Actually, why am I doing x? 134 00:06:20,029 --> 00:06:21,699 This would be the t-axis. 135 00:06:21,699 --> 00:06:22,939 We're doing the time domain. 136 00:06:22,939 --> 00:06:24,100 It could have been x. 137 00:06:24,100 --> 00:06:26,720 And then we'll call this f of t. 138 00:06:26,720 --> 00:06:29,590 So let me draw some arbitrary f of t. 139 00:06:29,589 --> 00:06:32,159 Let's say my function looks something crazy like that. 140 00:06:32,160 --> 00:06:35,100 141 00:06:35,100 --> 00:06:37,980 So this is my f of t. 142 00:06:37,980 --> 00:06:39,680 What if I'm modeling a physical system 143 00:06:39,680 --> 00:06:40,910 that doesn't do this? 144 00:06:40,910 --> 00:06:45,260 That actually at some point-- well, actually, let's say it 145 00:06:45,259 --> 00:06:47,329 stays at zero. 146 00:06:47,329 --> 00:06:51,050 It stays at zero until some value. 147 00:06:51,050 --> 00:06:53,405 Let's say it goes to zero until-- I don't know, I'll 148 00:06:53,404 --> 00:06:55,189 call that c again. 149 00:06:55,189 --> 00:06:57,870 And then at c, f of t kind of starts up. 150 00:06:57,870 --> 00:07:03,139 So right at c, f of t should start up, so it just kind of 151 00:07:03,139 --> 00:07:05,620 goes like this. 152 00:07:05,620 --> 00:07:09,269 So essentially what we have here is a combination of zero 153 00:07:09,269 --> 00:07:13,859 all the way, and then we have a shifted f of t. 154 00:07:13,860 --> 00:07:18,600 So at c, we have a shifted f of t, so it shifts that way. 155 00:07:18,600 --> 00:07:21,960 So how can we construct this yellow function, where it's 156 00:07:21,959 --> 00:07:25,149 essentially a shifted version of this green function, but 157 00:07:25,149 --> 00:07:26,959 it's zero below c? 158 00:07:26,959 --> 00:07:29,019 This green function might have continued. 159 00:07:29,019 --> 00:07:31,319 It might have gone something like this. 160 00:07:31,319 --> 00:07:34,324 It might have, continued and done something crazy, but what 161 00:07:34,324 --> 00:07:38,459 we did is we shifted it from here to there, and then we 162 00:07:38,459 --> 00:07:40,439 zeroed out everything before c. 163 00:07:40,439 --> 00:07:41,870 So how could we do that? 164 00:07:41,870 --> 00:07:44,879 Well, just shifting this function, you've learned in 165 00:07:44,879 --> 00:07:47,939 your Algebra II or your precalculus classes, to shift 166 00:07:47,939 --> 00:07:51,620 a function by c to the right, you just to replace your t 167 00:07:51,620 --> 00:07:52,970 with a t minus c. 168 00:07:52,970 --> 00:07:59,790 So this function right here is f of t minus c. 169 00:07:59,790 --> 00:08:02,960 And to make sure I get it right, what I always do is I 170 00:08:02,959 --> 00:08:06,269 imagine, OK, what's going to happen when t is equal to c? 171 00:08:06,269 --> 00:08:10,399 When t is equal to c, you're going to have a c minus a c, 172 00:08:10,399 --> 00:08:13,029 and you're going to have f of 0. 173 00:08:13,029 --> 00:08:15,839 So f of 0, it should be the same. 174 00:08:15,839 --> 00:08:19,269 So when t is equal to c, this value, the value of the 175 00:08:19,269 --> 00:08:22,560 function should be equivalent to the value of the original 176 00:08:22,560 --> 00:08:25,850 green function at zero, so it's equivalent to that value, 177 00:08:25,850 --> 00:08:27,110 which makes sense. 178 00:08:27,110 --> 00:08:31,110 If we go up one more above c, so let's say this is one more 179 00:08:31,110 --> 00:08:36,649 above c, so we get to this point, if t is c plus 1, then 180 00:08:36,649 --> 00:08:40,389 when you put c plus 1 minus c, you just have f of 1, and f of 181 00:08:40,389 --> 00:08:42,970 1 is really just this point right here. 182 00:08:42,970 --> 00:08:45,649 And so it'll be that f of 1, so it makes sense. 183 00:08:45,649 --> 00:08:48,389 So as we move one forward here, we're essentially at the 184 00:08:48,389 --> 00:08:52,350 same function value as we were there, so the shift works. 185 00:08:52,350 --> 00:08:55,320 But I said that we have to also-- if I just shifted this 186 00:08:55,320 --> 00:08:58,080 function, you would have all this other stuff, because you 187 00:08:58,080 --> 00:08:59,700 would have had all this other stuff when the function was 188 00:08:59,700 --> 00:09:02,500 back here still going on. 189 00:09:02,500 --> 00:09:03,769 The function-- I'll draw it 190 00:09:03,769 --> 00:09:05,809 lightly-- would still continue. 191 00:09:05,809 --> 00:09:08,329 But I said I wanted to zero out this function 192 00:09:08,330 --> 00:09:10,389 before we reach c. 193 00:09:10,389 --> 00:09:12,799 So how can I zero out that function? 194 00:09:12,799 --> 00:09:14,949 Well, I think it's pretty obvious to you. 195 00:09:14,950 --> 00:09:17,750 I started this video talking about the unit step function. 196 00:09:17,750 --> 00:09:19,679 So what if I multiply the unit step 197 00:09:19,679 --> 00:09:21,279 function times this thing? 198 00:09:21,279 --> 00:09:22,990 What's going to happen? 199 00:09:22,990 --> 00:09:26,870 So what if I-- my new function, I call it the unit 200 00:09:26,870 --> 00:09:35,129 step function up until c of t times f of t minus c? 201 00:09:35,129 --> 00:09:37,210 So what's going to happen? 202 00:09:37,210 --> 00:09:41,300 Until we get to c, the unit step function is zero when 203 00:09:41,299 --> 00:09:42,250 it's less than c. 204 00:09:42,250 --> 00:09:44,700 So you're going to have zero times I don't care what this 205 00:09:44,700 --> 00:09:48,070 is Zero times anything is zero, so this function is 206 00:09:48,070 --> 00:09:50,260 going to be zero. 207 00:09:50,259 --> 00:09:53,899 Once you hit c, the unit step function becomes 1. 208 00:09:53,899 --> 00:09:57,709 So once you pass c, this thing becomes a 1, and you're just 209 00:09:57,710 --> 00:09:59,320 left with 1 times your function. 210 00:09:59,320 --> 00:10:01,520 So then your function can behave as it would like to 211 00:10:01,519 --> 00:10:03,449 behave, and you actually shifted it. 212 00:10:03,450 --> 00:10:06,410 This t minus c is what actually shifted this green 213 00:10:06,409 --> 00:10:08,209 function over to the right. 214 00:10:08,210 --> 00:10:09,759 And this is actually going to be a very 215 00:10:09,759 --> 00:10:14,629 useful constructed function. 216 00:10:14,629 --> 00:10:17,110 And in a second, wer'e going to figure out the Laplace 217 00:10:17,110 --> 00:10:19,820 transform of this, and you're going to appreciate, I think, 218 00:10:19,820 --> 00:10:22,220 why this is a useful function to look at. 219 00:10:22,220 --> 00:10:26,800 But now you understand at least what it is and why it 220 00:10:26,799 --> 00:10:30,000 essentially shifts a function and zeroes out everything 221 00:10:30,000 --> 00:10:32,519 before that point. 222 00:10:32,519 --> 00:10:36,139 Well, I told you that this is a useful function, so we 223 00:10:36,139 --> 00:10:39,480 should add its Laplace transform to our library of 224 00:10:39,480 --> 00:10:42,720 Laplace transforms. So let's do that. 225 00:10:42,720 --> 00:10:49,410 Let's take a the Laplace transform of this, of the unit 226 00:10:49,409 --> 00:10:50,669 step function up to c. 227 00:10:50,669 --> 00:10:52,839 I'm doing it in fairly general terms. In the next video, 228 00:10:52,840 --> 00:10:56,200 we'll do a bunch of examples where we can apply this, but 229 00:10:56,200 --> 00:10:58,300 we should at least prove to ourselves what the Laplace 230 00:10:58,299 --> 00:11:01,000 transform of this thing is. 231 00:11:01,000 --> 00:11:04,399 Well, the Laplace transform of anything, or our definition of 232 00:11:04,399 --> 00:11:09,259 it so far, is the integral from 0 to infinity of e to the 233 00:11:09,259 --> 00:11:12,629 minus st times our function. 234 00:11:12,629 --> 00:11:18,240 So our function in this case is the unit step function, u 235 00:11:18,240 --> 00:11:24,950 sub c of t times f of t minus c dt. 236 00:11:24,950 --> 00:11:26,530 And this seems very general. 237 00:11:26,529 --> 00:11:29,669 It seems very hard to evaluate this integral at first, but 238 00:11:29,669 --> 00:11:32,539 maybe we can make some form of a substitution to get it into 239 00:11:32,539 --> 00:11:33,860 a term that we can appreciate. 240 00:11:33,860 --> 00:11:36,940 So let's make a substitution here. 241 00:11:36,940 --> 00:11:40,690 Let me pick a nice variable to work with. 242 00:11:40,690 --> 00:11:42,370 I don't know, we're not using an x anywhere. 243 00:11:42,370 --> 00:11:43,440 We might as well use an x. 244 00:11:43,440 --> 00:11:47,980 That's the most fun variable to work with. 245 00:11:47,980 --> 00:11:50,325 Sometimes, you'll see in a lot of math classes, they 246 00:11:50,325 --> 00:11:53,060 introduce these crazy Latin alphabets, and that by itself 247 00:11:53,059 --> 00:11:54,659 makes it hard to understand. 248 00:11:54,659 --> 00:11:59,509 So I like to stay away from those crazy Latin alphabets, 249 00:11:59,509 --> 00:12:01,179 so we'll just use a regular x. 250 00:12:01,179 --> 00:12:02,120 Let's make a substitute. 251 00:12:02,120 --> 00:12:08,789 Let's say that x is equal to t minus c. 252 00:12:08,789 --> 00:12:11,559 Or you could, if we added t to both sides, we could say that 253 00:12:11,559 --> 00:12:14,699 t is equal to x plus c. 254 00:12:14,700 --> 00:12:16,600 Let's see what happens to our subsitution. 255 00:12:16,600 --> 00:12:19,220 And also, if we took the derivative of both sides of 256 00:12:19,220 --> 00:12:22,360 this, or I guess the differential, you would get dx 257 00:12:22,360 --> 00:12:23,965 is equal to dt. 258 00:12:23,965 --> 00:12:27,460 Or I mean, if you took dx with respect to dt, you would get 259 00:12:27,460 --> 00:12:30,040 that to equal to 1. c is just a constant. 260 00:12:30,039 --> 00:12:32,899 Then if you multiply both sides by dt, you get dx is 261 00:12:32,899 --> 00:12:35,699 equal to dt, and that's a nice substitution. 262 00:12:35,700 --> 00:12:37,610 So what is our integral going to become with this 263 00:12:37,610 --> 00:12:39,100 substitution? 264 00:12:39,100 --> 00:12:42,670 So our integral this was t equals 0 to 265 00:12:42,669 --> 00:12:44,519 t is equal to infinity. 266 00:12:44,519 --> 00:12:51,879 When t is equal to 0, what is x going to be equal to? 267 00:12:51,879 --> 00:12:57,080 Well, x is going to be equal to minus c. 268 00:12:57,080 --> 00:13:00,170 Actually, before I go there, let me actually take a step 269 00:13:00,169 --> 00:13:03,349 back, because we could progress. 270 00:13:03,350 --> 00:13:04,720 We could go in this direction. 271 00:13:04,720 --> 00:13:09,149 But we could actually simplify it more before we do that. 272 00:13:09,149 --> 00:13:11,679 Let's go back to out original integral before we even made 273 00:13:11,679 --> 00:13:13,009 our substitution. 274 00:13:13,009 --> 00:13:15,679 If we're taking the integral from 0 to infinity of this 275 00:13:15,679 --> 00:13:18,289 thing, we already said what does this integral look like 276 00:13:18,289 --> 00:13:20,750 or what does this function look like? 277 00:13:20,750 --> 00:13:21,750 It's zero. 278 00:13:21,750 --> 00:13:26,870 We have this unit step function sitting right here. 279 00:13:26,870 --> 00:13:29,649 We have the unit step function sitting right there. 280 00:13:29,649 --> 00:13:31,980 So this whole expression is going to be zero 281 00:13:31,980 --> 00:13:33,980 until we get to c. 282 00:13:33,980 --> 00:13:36,220 This whole thing, by definition, this unit step 283 00:13:36,220 --> 00:13:38,399 function is zero until we get to c. 284 00:13:38,399 --> 00:13:40,399 So this everything's going to be zeroed out 285 00:13:40,399 --> 00:13:41,939 until we get to c. 286 00:13:41,940 --> 00:13:44,890 So we could essentially say, you know, we don't have to 287 00:13:44,889 --> 00:13:48,009 take the integral from t equals 0 to t equals infinity. 288 00:13:48,009 --> 00:13:52,539 We could take the integral-- let me write it here. 289 00:13:52,539 --> 00:13:54,449 I'll just use that old integral sign. 290 00:13:54,450 --> 00:13:59,070 We could just take the integral from t is equal to c 291 00:13:59,070 --> 00:14:05,790 to t is equal to infinity of e to the minus st, the unit step 292 00:14:05,789 --> 00:14:12,279 function, uc of t times f of t minus c dt. 293 00:14:12,279 --> 00:14:16,990 In fact, at this point, this unit step function, it has no 294 00:14:16,990 --> 00:14:18,279 use anymore. 295 00:14:18,279 --> 00:14:23,209 Because before t is equal to c, it's 0, and now that we're 296 00:14:23,210 --> 00:14:26,090 only worried about values above c, it's equal to 1, so 297 00:14:26,090 --> 00:14:28,440 it equals 1 in this context. 298 00:14:28,440 --> 00:14:30,240 I want to make that very clear to you. 299 00:14:30,240 --> 00:14:31,320 What did I do just here? 300 00:14:31,320 --> 00:14:35,340 I changed our bottom boundary from 0 to c. 301 00:14:35,340 --> 00:14:37,550 And I think you might realize why I did it when I was 302 00:14:37,549 --> 00:14:40,269 working with the substitution, because this will simplify 303 00:14:40,269 --> 00:14:42,090 things if we do this ahead of time. 304 00:14:42,090 --> 00:14:45,550 So if we have this unit step function, this thing is going 305 00:14:45,549 --> 00:14:49,959 to zero out this entire integral before we get to c. 306 00:14:49,960 --> 00:14:51,950 Remember, this definite integral is really just the 307 00:14:51,950 --> 00:14:56,060 area under this curve of this whole function, of the unit 308 00:14:56,059 --> 00:14:57,979 step function times all of this stuff. 309 00:14:57,980 --> 00:15:00,200 All of this stuff, when we multiply it, is going to be 310 00:15:00,200 --> 00:15:03,050 zero until we get to some value c. 311 00:15:03,049 --> 00:15:05,620 And then above c, it's going to be e to the minus st times 312 00:15:05,620 --> 00:15:06,759 f of t minus c. 313 00:15:06,759 --> 00:15:08,819 So it's going to start doing all this crazy stuff. 314 00:15:08,820 --> 00:15:12,610 So if we want to essentially find the area under this 315 00:15:12,610 --> 00:15:16,840 curve, we can ignore all the stuff that happens before c. 316 00:15:16,840 --> 00:15:19,519 So instead of going from t equals 0 to infinity, we can 317 00:15:19,519 --> 00:15:22,639 go from t is equal to c to infinity because there was no 318 00:15:22,639 --> 00:15:25,539 area before t was equal to c. 319 00:15:25,539 --> 00:15:26,959 So that's all I did here. 320 00:15:26,960 --> 00:15:28,920 And then the other thing I said is that the unit step 321 00:15:28,919 --> 00:15:34,639 function, it's going to be 1 over this entire range of 322 00:15:34,639 --> 00:15:38,419 potential t-values, so we can just kind of ignore it. 323 00:15:38,419 --> 00:15:41,370 It's just going to be 1 this entire time, so our integral 324 00:15:41,370 --> 00:15:46,379 simplifies to the definite integral from t is equal to c 325 00:15:46,379 --> 00:15:53,439 to t is equal to infinity of e to the minus st times f of t 326 00:15:53,440 --> 00:15:56,025 minus is c dt. 327 00:15:56,024 --> 00:15:58,029 And this will simplify it a good bit. 328 00:15:58,029 --> 00:15:59,600 I was going down the other road when I did the 329 00:15:59,600 --> 00:16:01,830 substitution first, which would have worked, but I think 330 00:16:01,830 --> 00:16:04,750 the argument as to why I could have changed the boundaries 331 00:16:04,750 --> 00:16:06,759 would've been a harder argument to make. 332 00:16:06,759 --> 00:16:09,899 So now that we had this, let's go back and make that 333 00:16:09,899 --> 00:16:13,759 substitution that x is equal to t minus c. 334 00:16:13,759 --> 00:16:17,340 So our integral becomes-- I'll do it in green-- when t is 335 00:16:17,340 --> 00:16:18,990 equal to c, what is x? 336 00:16:18,990 --> 00:16:20,950 Then x is 0, right? 337 00:16:20,950 --> 00:16:22,440 c minus c is 0. 338 00:16:22,440 --> 00:16:24,680 When t is equal to infinity, what is x? 339 00:16:24,679 --> 00:16:26,719 Well x is, you know, infinity minus any constant is still 340 00:16:26,720 --> 00:16:29,460 going to be infinity, or if the limit is t approaches 341 00:16:29,460 --> 00:16:32,920 infinity, x is still going to be infinity here. 342 00:16:32,919 --> 00:16:37,099 And it's the integral of e to the minus s, but now instead 343 00:16:37,100 --> 00:16:39,245 of a t, we have the substitution. 344 00:16:39,245 --> 00:16:42,529 If we said x is equal to t minus c, then we can just add 345 00:16:42,529 --> 00:16:45,659 c to both sides and get t is equal to x plus c. 346 00:16:45,659 --> 00:16:53,829 So you get x plus c there, and then times the function f of t 347 00:16:53,830 --> 00:16:58,230 minus c, but we said t minus c is the same thing as x. 348 00:16:58,230 --> 00:17:00,370 And dt is the same thing is dx. 349 00:17:00,370 --> 00:17:05,460 I showed you that right there, so we can write this as dx. 350 00:17:05,460 --> 00:17:08,700 Now this is starting to look a little bit interesting. 351 00:17:08,700 --> 00:17:10,779 So what is this equal to? 352 00:17:10,779 --> 00:17:14,730 This is equal to the integral from 0 to infinity-- let me 353 00:17:14,730 --> 00:17:23,578 expand this out-- of e to the minus sx minus sc 354 00:17:23,578 --> 00:17:28,240 times f of x dx. 355 00:17:28,240 --> 00:17:30,359 Now, what is the equal to? 356 00:17:30,359 --> 00:17:34,500 Well, we could factor out an e to the minus sc and bring it 357 00:17:34,500 --> 00:17:36,299 outside of the integral, because this has nothing to do 358 00:17:36,299 --> 00:17:39,149 with what we're taking the integral with respect to. 359 00:17:39,150 --> 00:17:39,970 So let's do that. 360 00:17:39,970 --> 00:17:44,529 Let me take this guy out, and maybe just to not confuse you, 361 00:17:44,529 --> 00:17:45,599 let me rewrite the whole thing. 362 00:17:45,599 --> 00:17:46,649 0 to infinity. 363 00:17:46,650 --> 00:17:50,009 I could rewrite this e term as e-- actually, let me 364 00:17:50,009 --> 00:17:51,460 write it this way. 365 00:17:51,460 --> 00:17:55,930 I'll do what was already in green as e to the minus sx 366 00:17:55,930 --> 00:18:00,820 times e to the minus sc. 367 00:18:00,819 --> 00:18:01,859 Common base. 368 00:18:01,859 --> 00:18:05,799 So if I were to multiply these two, I could just add the 369 00:18:05,799 --> 00:18:10,430 exponents, which you would get that up there, times 370 00:18:10,430 --> 00:18:14,580 f of x, d of x. 371 00:18:14,579 --> 00:18:17,199 This is a constant term with respect to x, so we can just 372 00:18:17,200 --> 00:18:18,720 factor it out. 373 00:18:18,720 --> 00:18:22,350 We can just factor this thing out right there, so then you 374 00:18:22,349 --> 00:18:28,289 get e to the minus sc times the integral from 0 to 375 00:18:28,289 --> 00:18:37,809 infinity of e to the minus sx times f of x dx. 376 00:18:37,809 --> 00:18:40,089 Now, what were we doing here the whole time? 377 00:18:40,089 --> 00:18:49,069 We were taking the Laplace transform of the unit step 378 00:18:49,069 --> 00:18:51,950 function that goes up to c, and then it's 0 up to c, and 379 00:18:51,950 --> 00:18:58,000 it's 1 after that, of t times some shifted function 380 00:18:58,000 --> 00:19:01,680 f of t minus c. 381 00:19:01,680 --> 00:19:05,720 And now we got that as being equal to this thing, and we 382 00:19:05,720 --> 00:19:06,710 made a substitution. 383 00:19:06,710 --> 00:19:08,490 We simplified it a little bit. 384 00:19:08,490 --> 00:19:16,609 e to the minus sc times the integral from 0 to infinity of 385 00:19:16,609 --> 00:19:21,599 e to the minus sx f of x dx. 386 00:19:21,599 --> 00:19:24,179 Something about the tablet doesn't work properly right 387 00:19:24,180 --> 00:19:26,289 around this period. 388 00:19:26,289 --> 00:19:30,920 But this should look interesting to you. 389 00:19:30,920 --> 00:19:32,600 What is this? 390 00:19:32,599 --> 00:19:35,469 This is the Laplace transform of f of x. 391 00:19:35,470 --> 00:19:36,279 Let me write that down. 392 00:19:36,279 --> 00:19:40,609 What's the Laplace transform of-- well, I could write it as 393 00:19:40,609 --> 00:19:42,639 f of t or f of x. 394 00:19:42,640 --> 00:19:46,910 The Laplace transform of f of t is equal to the integral 395 00:19:46,910 --> 00:19:55,680 from 0 to infinity of e to the minus st times f of t dt. 396 00:19:55,680 --> 00:19:58,039 This and this are the exact same thing. 397 00:19:58,039 --> 00:19:59,519 We're just using a t here. 398 00:19:59,519 --> 00:20:00,960 We're using an x here. 399 00:20:00,960 --> 00:20:01,789 No difference. 400 00:20:01,789 --> 00:20:03,909 They're just letters. 401 00:20:03,910 --> 00:20:06,490 This is f of t. 402 00:20:06,490 --> 00:20:08,870 e to the minus st times f of t dt. 403 00:20:08,869 --> 00:20:11,269 I could have also rewritten it as the Laplace 404 00:20:11,269 --> 00:20:14,970 transform of f of t. 405 00:20:14,970 --> 00:20:17,789 I could write this as the integral from 0 to infinity of 406 00:20:17,789 --> 00:20:24,470 e to the minus sy times f of y dy. 407 00:20:24,470 --> 00:20:25,930 I could do it by anything because this 408 00:20:25,930 --> 00:20:26,740 is a definite integral. 409 00:20:26,740 --> 00:20:28,329 The y's are going to disappear, 410 00:20:28,329 --> 00:20:29,189 and we've seen that. 411 00:20:29,190 --> 00:20:32,500 All you're left with is a function of s. 412 00:20:32,500 --> 00:20:35,910 This ends up being some capital, well, you know, we 413 00:20:35,910 --> 00:20:39,140 could write some capital function of s. 414 00:20:39,140 --> 00:20:40,060 So this is interesting. 415 00:20:40,059 --> 00:20:44,679 This is the Laplace transform of f of t times some scaling 416 00:20:44,680 --> 00:20:48,060 factor, and that's what we set out to show. 417 00:20:48,059 --> 00:20:54,789 So we can now show that the Laplace transform of the unit 418 00:20:54,789 --> 00:21:02,700 step function times some function t minus c is equal to 419 00:21:02,700 --> 00:21:07,870 this function right here, e to the minus sc, where this c is 420 00:21:07,869 --> 00:21:11,649 the same as this c right here, times the Laplace 421 00:21:11,650 --> 00:21:13,740 transform of f of t. 422 00:21:13,740 --> 00:21:17,370 Times the Laplace transform-- I don't know what's going on 423 00:21:17,369 --> 00:21:21,449 with the tablet right there-- of f of t. 424 00:21:21,450 --> 00:21:22,100 Let me write that. 425 00:21:22,099 --> 00:21:25,449 This is equal to-- because it's looking funny there-- e 426 00:21:25,450 --> 00:21:32,539 to the minus sc times the Laplace transform of f of t. 427 00:21:32,539 --> 00:21:33,865 So this is our result. 428 00:21:33,865 --> 00:21:37,660 429 00:21:37,660 --> 00:21:39,740 Now, what does this mean? 430 00:21:39,740 --> 00:21:42,289 Oh, look it back-filled it somehow. 431 00:21:42,289 --> 00:21:43,180 What does this mean? 432 00:21:43,180 --> 00:21:45,285 What can we do with this? 433 00:21:45,285 --> 00:21:50,420 Well, let's say we wanted to figure out the Laplace 434 00:21:50,420 --> 00:21:53,910 transform of the unit step function that 435 00:21:53,910 --> 00:21:56,200 starts up at pi of t. 436 00:21:56,200 --> 00:21:59,430 And let's say we're taking something that we know well: 437 00:21:59,430 --> 00:22:05,509 sine of t minus pi. 438 00:22:05,509 --> 00:22:08,569 So we shifted it, right? 439 00:22:08,569 --> 00:22:10,750 This thing is really malfunctioning at this point 440 00:22:10,750 --> 00:22:11,450 right here. 441 00:22:11,450 --> 00:22:14,600 Let me pause it. 442 00:22:14,599 --> 00:22:15,439 I just paused. 443 00:22:15,440 --> 00:22:16,890 Sorry if that was a little disconcerting. 444 00:22:16,890 --> 00:22:19,420 I just paused the video because it was having trouble 445 00:22:19,420 --> 00:22:21,950 recording at some point on my little board. 446 00:22:21,950 --> 00:22:25,130 So let me rewrite the result that we proved just now. 447 00:22:25,130 --> 00:22:29,430 We showed that the Laplace transform of the unit step 448 00:22:29,430 --> 00:22:34,519 function t, and it goes to 1 at some value c times some 449 00:22:34,519 --> 00:22:38,819 function that's shifted by c to the right. 450 00:22:38,819 --> 00:22:43,769 It's equal to e to the minus cs times the Laplace transform 451 00:22:43,769 --> 00:22:45,519 of just the unshifted function. 452 00:22:45,519 --> 00:22:46,400 That was our result. 453 00:22:46,400 --> 00:22:48,630 That was the big takeaway from this video. 454 00:22:48,630 --> 00:22:51,550 And if this seems like some Byzantine, hard-to-understand 455 00:22:51,549 --> 00:22:53,629 result, we can apply it. 456 00:22:53,630 --> 00:22:55,930 So let's say the Laplace transform, this is what I was 457 00:22:55,930 --> 00:23:01,080 doing right before the actual pen tablet started 458 00:23:01,079 --> 00:23:01,919 malfunctioning. 459 00:23:01,920 --> 00:23:04,500 If we want to take the Laplace transform of the unit step 460 00:23:04,500 --> 00:23:09,609 function that goes to 1 at pi, t times the sine function 461 00:23:09,609 --> 00:23:15,219 shifted by pi to the right, we know that this is going to be 462 00:23:15,220 --> 00:23:19,319 equal to e to the minus cs. 463 00:23:19,319 --> 00:23:24,700 c is pi in this case, so minus pi s times the Laplace 464 00:23:24,700 --> 00:23:27,759 transform of the unshifted function. 465 00:23:27,759 --> 00:23:29,029 So in this case, it's the Laplace 466 00:23:29,029 --> 00:23:32,299 transform of sine of t. 467 00:23:32,299 --> 00:23:36,460 And we know what the Laplace transform of sine of t is. 468 00:23:36,460 --> 00:23:41,250 It's just 1 over s squared plus 1. 469 00:23:41,250 --> 00:23:43,769 So the Laplace transform of this thing here, which before 470 00:23:43,769 --> 00:23:47,720 this video seemed like something crazy, we now know 471 00:23:47,720 --> 00:23:49,089 is this times this. 472 00:23:49,089 --> 00:23:53,379 So it's e to the minus pi s times this, or we could just 473 00:23:53,380 --> 00:23:58,900 write it as e to the minus pi s over s squared plus 1. 474 00:23:58,900 --> 00:24:00,990 We'll do a couple more examples of this in the next 475 00:24:00,990 --> 00:24:02,900 video, where we go back and forth between the Laplace 476 00:24:02,900 --> 00:24:05,950 world and the t and between the s domain 477 00:24:05,950 --> 00:24:06,809 and the time domain. 478 00:24:06,809 --> 00:24:10,730 And I'll show you how this is a very useful result to take a 479 00:24:10,730 --> 00:24:13,910 lot of Laplace transforms and to invert a lot of Laplace 480 00:24:13,910 --> 00:24:15,160 transforms.