1 00:00:00,000 --> 00:00:00,790 2 00:00:00,790 --> 00:00:01,530 Welcome back. 3 00:00:01,530 --> 00:00:04,139 We're finally using the Laplace Transform to do 4 00:00:04,139 --> 00:00:05,459 something useful. 5 00:00:05,459 --> 00:00:07,879 In the first part of this problem, we just had this 6 00:00:07,879 --> 00:00:09,849 fairly straightforward differential equation. 7 00:00:09,849 --> 00:00:11,730 And I know it's a little bit frustrating right now, because 8 00:00:11,730 --> 00:00:13,650 you're like, this is such an easy one to solve using the 9 00:00:13,650 --> 00:00:15,259 characteristic equation. 10 00:00:15,259 --> 00:00:16,640 Why are we doing Laplace Transforms? 11 00:00:16,640 --> 00:00:18,410 Well I just want to show you that they can solve even these 12 00:00:18,410 --> 00:00:20,660 problems. But later on there are going to be classes of 13 00:00:20,660 --> 00:00:24,649 problems that, frankly, our traditional methods aren't as 14 00:00:24,649 --> 00:00:26,500 good as the Laplace Transform. 15 00:00:26,500 --> 00:00:27,789 But anyway, how did we solve this? 16 00:00:27,789 --> 00:00:29,849 We just took the Laplace Transform of both sides of 17 00:00:29,850 --> 00:00:30,870 this equation. 18 00:00:30,870 --> 00:00:33,539 We got all of this hairy mess. 19 00:00:33,539 --> 00:00:36,144 We used the property of the derivative of functions, where 20 00:00:36,145 --> 00:00:38,030 you take the Laplace Transform, and we ended up, 21 00:00:38,030 --> 00:00:40,100 after doing a lot of algebra essentially, we got this. 22 00:00:40,100 --> 00:00:43,109 We got the Laplace Transform of y is equal to this thing. 23 00:00:43,109 --> 00:00:45,329 We just took the Laplace Transform of both sides and 24 00:00:45,329 --> 00:00:47,170 manipulated algebraically. 25 00:00:47,170 --> 00:00:50,539 So now our task in this video is to figure out what y's 26 00:00:50,539 --> 00:00:52,786 Laplace Transform is this thing? 27 00:00:52,786 --> 00:00:55,070 And essentially what we're trying to do, is we're trying 28 00:00:55,070 --> 00:00:58,250 to take the inverse Laplace Transform of both sides of 29 00:00:58,250 --> 00:00:58,810 this equation. 30 00:00:58,810 --> 00:01:03,010 So another way to say it, we could say that y-- if we take 31 00:01:03,009 --> 00:01:05,246 the inverse Laplace Transform of both sides-- we could say 32 00:01:05,246 --> 00:01:09,189 that y is equal to the inverse Laplace 33 00:01:09,189 --> 00:01:12,259 Transform of this thing. 34 00:01:12,260 --> 00:01:19,371 2s plus 13, over s squared plus 5s plus 6. 35 00:01:19,371 --> 00:01:21,780 Now we'll eventually actually learn the formal definition of 36 00:01:21,780 --> 00:01:23,599 the inverse Laplace Transform. 37 00:01:23,599 --> 00:01:27,039 How do you go from the s domain to the t domain? 38 00:01:27,040 --> 00:01:28,210 Or how do you go from the frequency 39 00:01:28,209 --> 00:01:29,879 domain to the time domain? 40 00:01:29,879 --> 00:01:32,009 We're not going to worry about that right now. 41 00:01:32,010 --> 00:01:34,329 What we're going to do is we're going to get this into a 42 00:01:34,329 --> 00:01:36,750 form that we recognize, and say, oh, 43 00:01:36,750 --> 00:01:38,180 I know those functions. 44 00:01:38,180 --> 00:01:40,200 That's the Laplace Transform of whatever and whatever. 45 00:01:40,200 --> 00:01:42,269 And then we'll know what y is. 46 00:01:42,269 --> 00:01:44,229 So let's try to do that. 47 00:01:44,230 --> 00:01:47,750 So what we're going to use is something that you probably 48 00:01:47,750 --> 00:01:50,969 haven't used since Algebra two, which is I think when 49 00:01:50,969 --> 00:01:52,719 it's taught in, you know, eighth, ninth, 50 00:01:52,719 --> 00:01:53,679 or 10th grade, depending. 51 00:01:53,680 --> 00:01:55,740 And you finally see it now in differential equations that it 52 00:01:55,739 --> 00:01:57,640 actually has some use. 53 00:01:57,640 --> 00:01:58,450 Let me write it. 54 00:01:58,450 --> 00:02:01,780 We're going to use partial fraction expansion. 55 00:02:01,780 --> 00:02:04,079 And I'll do a little primer on that, in case you don't 56 00:02:04,079 --> 00:02:04,739 remember it. 57 00:02:04,739 --> 00:02:08,538 So anyway, let's just factor the bottom part right here. 58 00:02:08,538 --> 00:02:11,509 And you'll see where I'm going with this. 59 00:02:11,509 --> 00:02:19,129 So if I factor the bottom, I get s plus 2 times s plus 3. 60 00:02:19,129 --> 00:02:25,419 And what we want to do, is we want to rewrite this fraction 61 00:02:25,419 --> 00:02:28,979 as the sum of 2-- I guess you could 62 00:02:28,979 --> 00:02:30,569 call it partial fractions. 63 00:02:30,569 --> 00:02:33,229 I think that's why it's called partial fraction expansion. 64 00:02:33,229 --> 00:02:41,109 So we want to write this as a sum of A over s plus 2, plus B 65 00:02:41,110 --> 00:02:43,250 over s plus 3. 66 00:02:43,250 --> 00:02:48,280 And if we can do this, then-- and bells might already be 67 00:02:48,280 --> 00:02:54,710 ringing in your head-- we know that these things that look 68 00:02:54,710 --> 00:02:56,650 like this are the Laplace Transform of functions that 69 00:02:56,650 --> 00:02:57,960 we've already solved for. 70 00:02:57,960 --> 00:03:01,110 And I'll do a little review on that in a second. 71 00:03:01,110 --> 00:03:03,680 But anyway, how do we figure out A and B? 72 00:03:03,680 --> 00:03:06,800 Well if we were to actually add A and B, if we were to-- 73 00:03:06,800 --> 00:03:12,850 let's do a little aside right here-- so if we said that A-- 74 00:03:12,849 --> 00:03:16,479 so if we were to give them a common denominator, which is 75 00:03:16,479 --> 00:03:21,294 this, s plus 2 times s plus 3. 76 00:03:21,294 --> 00:03:22,750 Then what would A become? 77 00:03:22,750 --> 00:03:25,349 We'd have to multiply A times s plus 3, right? 78 00:03:25,349 --> 00:03:29,030 So we'd get As plus 3A. 79 00:03:29,030 --> 00:03:31,640 80 00:03:31,639 --> 00:03:34,219 This, as I've written it right now, is the same thing as A 81 00:03:34,219 --> 00:03:34,979 over s plus 2. 82 00:03:34,979 --> 00:03:39,079 You could cancel out an s plus 3 in the top and the bottom. 83 00:03:39,080 --> 00:03:41,080 And now we're going to add the B to it. 84 00:03:41,080 --> 00:03:45,500 So plus-- I'll do that in a different color-- plus-- well, 85 00:03:45,500 --> 00:03:48,110 if we have this as the denominator, we could multiply 86 00:03:48,110 --> 00:03:49,140 the numerator and the denominator 87 00:03:49,139 --> 00:03:51,099 by s plus 2, right? 88 00:03:51,099 --> 00:03:58,889 To get B times s, plus 2B, and that's going 89 00:03:58,889 --> 00:04:01,469 to equal this thing. 90 00:04:01,469 --> 00:04:03,359 And all I did is I added these two fractions. 91 00:04:03,360 --> 00:04:04,350 Nothing fancier than there. 92 00:04:04,349 --> 00:04:05,919 That was Algebra two. 93 00:04:05,919 --> 00:04:07,229 Actually, I think I should do an actual 94 00:04:07,229 --> 00:04:08,334 video on that as well. 95 00:04:08,335 --> 00:04:11,409 But that's going to equal this thing. 96 00:04:11,409 --> 00:04:20,899 2s plus 13, all of that over s plus 2 times s plus 3. 97 00:04:20,899 --> 00:04:23,250 Notice in all differential equations, the hairiest part's 98 00:04:23,250 --> 00:04:24,689 always the algebra. 99 00:04:24,689 --> 00:04:26,779 So now what we do is we match up. 100 00:04:26,779 --> 00:04:29,109 We say, well, let's add the s terms here. 101 00:04:29,110 --> 00:04:31,439 And we could say that the numerators have to equal each 102 00:04:31,439 --> 00:04:33,350 other, because the denominators are equal. 103 00:04:33,350 --> 00:04:51,960 So we have A plus Bs plus 3A plus 2B is equal to 2s plus B. 104 00:04:51,959 --> 00:04:55,109 So the coefficient on s, on the right-hand side, is 2. 105 00:04:55,110 --> 00:04:57,910 The coefficient on the left-hand side is A plus B, so 106 00:04:57,910 --> 00:04:59,980 we know that A plus B is equal to 2. 107 00:04:59,980 --> 00:05:02,950 108 00:05:02,949 --> 00:05:08,269 And then on the right-hand side, we see 3A plus 2B must 109 00:05:08,269 --> 00:05:11,149 be equal to-- oh, this is a 13. 110 00:05:11,149 --> 00:05:11,929 Did I say B? 111 00:05:11,930 --> 00:05:14,189 This is a 13. 112 00:05:14,189 --> 00:05:15,529 That's a 13. 113 00:05:15,529 --> 00:05:16,889 It looks just like a B, right? 114 00:05:16,889 --> 00:05:19,099 That was 2s plus 13. 115 00:05:19,100 --> 00:05:28,640 Anyway, so on the right-hand side I get, it was 3A plus 2B 116 00:05:28,639 --> 00:05:32,149 is equal to 13. 117 00:05:32,149 --> 00:05:34,789 Now we have two equations with two unknowns, 118 00:05:34,790 --> 00:05:35,610 and what do we get? 119 00:05:35,610 --> 00:05:37,250 I know this is very tiresome, but it'll be 120 00:05:37,250 --> 00:05:38,259 satisfying in the end. 121 00:05:38,259 --> 00:05:39,459 Because you'll actually solve something 122 00:05:39,459 --> 00:05:40,514 with the Laplace Transform. 123 00:05:40,514 --> 00:05:43,659 So let's multiply the top equation by 2, or let's just 124 00:05:43,660 --> 00:05:44,350 say minus 2. 125 00:05:44,350 --> 00:05:50,320 So we get minus 2A minus 2B equals minus 4. 126 00:05:50,319 --> 00:05:54,240 And then we get-- add the two equations-- you get A is equal 127 00:05:54,240 --> 00:05:57,170 to-- these cancel out-- A is equal to 9. 128 00:05:57,170 --> 00:05:58,000 Great. 129 00:05:58,000 --> 00:06:00,529 If A is equal to 9, what is B equal to? 130 00:06:00,529 --> 00:06:05,519 B is equal to 9 plus what is equal to 2? 131 00:06:05,519 --> 00:06:09,129 Or 2 minus 9 is minus 7. 132 00:06:09,129 --> 00:06:12,159 And we have done some serious simplification. 133 00:06:12,160 --> 00:06:15,770 Because now we can rewrite this whole expression as the 134 00:06:15,769 --> 00:06:23,389 Laplace Transform of y is equal to A over s plus 2, is 135 00:06:23,389 --> 00:06:34,060 equal to 9 over s plus 2, minus 7 over s plus 3. 136 00:06:34,060 --> 00:06:38,810 Or another way of writing it, we could write it as equal to 137 00:06:38,810 --> 00:06:47,620 9 times 1 over s plus 2, minus 7 times 1 over s plus 3. 138 00:06:47,620 --> 00:06:49,629 Why did I take the trouble to do this? 139 00:06:49,629 --> 00:06:51,920 Well hopefully, you'll recognize this was actually 140 00:06:51,920 --> 00:06:55,009 the second Laplace Transform we figured out. 141 00:06:55,009 --> 00:06:58,199 142 00:06:58,199 --> 00:06:59,289 What was that? 143 00:06:59,290 --> 00:07:01,950 I'll write it down here just so you remember it. 144 00:07:01,949 --> 00:07:11,969 It was the Laplace Transform of e to the at, was equal to 1 145 00:07:11,970 --> 00:07:15,000 over s minus a. 146 00:07:15,000 --> 00:07:18,069 That was the second Laplace Transform we figured out. 147 00:07:18,069 --> 00:07:21,199 So this is interesting. 148 00:07:21,199 --> 00:07:22,879 This is the Laplace Transform of what? 149 00:07:22,879 --> 00:07:24,730 So if we were to take the inverse Laplace Transform-- 150 00:07:24,730 --> 00:07:26,900 actually let me just stay consistent. 151 00:07:26,899 --> 00:07:32,819 So that means that this is the Laplace Transform of y, is 152 00:07:32,819 --> 00:07:36,360 equal to 9 times the Laplace Transform of what? 153 00:07:36,360 --> 00:07:38,850 If we just do pattern matching, if this is s minus 154 00:07:38,850 --> 00:07:41,080 a, then a is minus 2. 155 00:07:41,079 --> 00:07:45,219 So 9 times the Laplace Transform of e 156 00:07:45,220 --> 00:07:48,970 to the minus 2t. 157 00:07:48,970 --> 00:07:50,380 Does that make sense? 158 00:07:50,379 --> 00:07:52,899 Take this, put it in this one, which we figured out, and you 159 00:07:52,899 --> 00:07:54,189 get 1 over s plus 2. 160 00:07:54,189 --> 00:07:56,204 And let me clean this up a little bit, because I'm going 161 00:07:56,204 --> 00:07:57,454 to need that real estate. 162 00:07:57,454 --> 00:08:02,449 163 00:08:02,449 --> 00:08:03,120 I'll write this. 164 00:08:03,120 --> 00:08:06,269 I'll leave that there, because we'll still use that. 165 00:08:06,269 --> 00:08:10,799 And then we have minus 7 times-- this is the Laplace 166 00:08:10,800 --> 00:08:11,500 Transform of what? 167 00:08:11,500 --> 00:08:16,189 This is the Laplace Transform of e to the minus 3t. 168 00:08:16,189 --> 00:08:20,449 169 00:08:20,449 --> 00:08:25,189 This pattern matching, you're like, wow, if you saw this, 170 00:08:25,189 --> 00:08:27,269 you would go to your Laplace Transform table, if you didn't 171 00:08:27,269 --> 00:08:28,639 remember it, you'd see this. 172 00:08:28,639 --> 00:08:31,430 You're like, wow, that looks a lot like that. 173 00:08:31,430 --> 00:08:33,000 I just have to figure out what a is. 174 00:08:33,000 --> 00:08:33,960 I have s plus 3. 175 00:08:33,960 --> 00:08:35,320 I have s minus a. 176 00:08:35,320 --> 00:08:37,520 So in this case, a is equal to minus 3. 177 00:08:37,519 --> 00:08:39,894 So if a is equal to minus 3, this is the Laplace Transform 178 00:08:39,894 --> 00:08:42,759 of e to the minus 3t. 179 00:08:42,759 --> 00:08:46,389 So now we can take the inverse Laplace-- actually, 180 00:08:46,389 --> 00:08:47,069 before we do that. 181 00:08:47,070 --> 00:08:49,750 We know that because the Laplace Transform is a linear 182 00:08:49,750 --> 00:08:54,759 operator-- and actually now I can delete this down here-- we 183 00:08:54,759 --> 00:08:57,139 know that the Laplace Transform is a linear 184 00:08:57,139 --> 00:09:00,110 operator, so we can write this. 185 00:09:00,110 --> 00:09:01,710 And you normally wouldn't go through all of these steps. 186 00:09:01,710 --> 00:09:05,530 I just really want to make you understand what we're doing. 187 00:09:05,529 --> 00:09:08,139 So we could say that this is the same thing as the Laplace 188 00:09:08,139 --> 00:09:16,699 Transform of 9e to the minus 2t, minus 7e to the minus 3t. 189 00:09:16,700 --> 00:09:19,570 190 00:09:19,570 --> 00:09:20,860 Now we have something interesting. 191 00:09:20,860 --> 00:09:23,399 The Laplace Transform of y is equal to the Laplace 192 00:09:23,399 --> 00:09:24,829 Transform of this. 193 00:09:24,830 --> 00:09:30,570 Well if that's the case, then y must be equal to 9e to the 194 00:09:30,570 --> 00:09:35,320 minus 2t, minus 7e to the minus 3t. 195 00:09:35,320 --> 00:09:38,080 And I never proved to you, but the Laplace Transform is 196 00:09:38,080 --> 00:09:40,300 actually a 1:1 Transformation. 197 00:09:40,299 --> 00:09:43,329 That if a function's Laplace Transform, if I take a 198 00:09:43,330 --> 00:09:45,420 function against the Laplace Transform, and then if I were 199 00:09:45,419 --> 00:09:48,809 take the inverse Laplace Transform, the only function 200 00:09:48,809 --> 00:09:50,819 whose Laplace Transform that that is, is 201 00:09:50,820 --> 00:09:51,710 that original function. 202 00:09:51,710 --> 00:09:54,550 It's not like two different functions can have the same 203 00:09:54,549 --> 00:09:56,049 Laplace Transform. 204 00:09:56,049 --> 00:09:58,679 Anyway, a couple of things to think about here. 205 00:09:58,679 --> 00:10:02,069 Notice, we had that thing that kind of looked like a 206 00:10:02,070 --> 00:10:04,980 characteristic equation pop up here and there. 207 00:10:04,980 --> 00:10:08,340 And we still have to solve a system of two equations with 208 00:10:08,340 --> 00:10:08,850 two unknowns. 209 00:10:08,850 --> 00:10:13,750 Those are both things that we had to do when we solve an 210 00:10:13,750 --> 00:10:17,029 initial value problem, when we use just traditional, the 211 00:10:17,029 --> 00:10:18,339 characteristic equation. 212 00:10:18,340 --> 00:10:20,149 But here it happened all at once. 213 00:10:20,149 --> 00:10:21,899 And frankly it was a little bit hairier because we had to 214 00:10:21,899 --> 00:10:24,039 do all this partial fraction expansion. 215 00:10:24,039 --> 00:10:25,049 But it's pretty neat. 216 00:10:25,049 --> 00:10:28,209 The Laplace Transform got us something useful. 217 00:10:28,210 --> 00:10:30,970 In the next video I'll actually do a non-homogeneous 218 00:10:30,970 --> 00:10:33,850 equation, and show you that the Laplace Transform applies 219 00:10:33,850 --> 00:10:34,830 equally well there. 220 00:10:34,830 --> 00:10:37,680 So it's kind of a more consistent theory of solving 221 00:10:37,679 --> 00:10:39,729 differential equations, instead of kind of guessing 222 00:10:39,730 --> 00:10:43,139 solutions, and solving for coefficients and all of that. 223 00:10:43,139 --> 00:10:45,100 See you in the next video. 224 00:10:45,100 --> 00:10:45,399