1 00:00:00,000 --> 00:00:00,710 2 00:00:00,710 --> 00:00:03,149 Let's apply everything we've learned to an actual 3 00:00:03,149 --> 00:00:04,389 differential equation. 4 00:00:04,389 --> 00:00:07,849 Instead of just taking Laplace transforms and taking their 5 00:00:07,849 --> 00:00:09,939 inverse, let's actually solve a problem. 6 00:00:09,939 --> 00:00:13,879 So let's say that I have the second derivative of my 7 00:00:13,880 --> 00:00:22,550 function y plus 4 times my function y is equal to sine of 8 00:00:22,550 --> 00:00:36,370 t minus the unit step function 0 up until 2 pi of t times 9 00:00:36,369 --> 00:00:42,890 sine of t minus 2 pi. 10 00:00:42,890 --> 00:00:45,119 Let's solve this differential equation, an 11 00:00:45,119 --> 00:00:45,869 interpretation of it. 12 00:00:45,869 --> 00:00:47,949 And I actually do a whole playlist on interpretations of 13 00:00:47,950 --> 00:00:50,440 differential equations and how you model it, but you know, 14 00:00:50,439 --> 00:00:52,489 you can kind of view this is a forcing function. 15 00:00:52,490 --> 00:00:55,829 That it's a weird forcing function of this being applied 16 00:00:55,829 --> 00:00:58,280 to some weight with, you know, this is the 17 00:00:58,280 --> 00:00:59,469 acceleration term, right? 18 00:00:59,469 --> 00:01:01,490 The second derivative with respect to time is the 19 00:01:01,490 --> 00:01:02,289 acceleration. 20 00:01:02,289 --> 00:01:09,539 So the mass would be 1 whatever units, and then as a 21 00:01:09,540 --> 00:01:11,340 function of its position, this is probably some type of 22 00:01:11,340 --> 00:01:12,150 spring constant. 23 00:01:12,150 --> 00:01:13,210 Anyway, I won't go there. 24 00:01:13,209 --> 00:01:15,649 I don't want to waste your time with the interpretation 25 00:01:15,650 --> 00:01:17,859 of it, but let's solve it. 26 00:01:17,859 --> 00:01:19,599 We can do more about interpretations later. 27 00:01:19,599 --> 00:01:21,599 So we're going take the Laplace transform of both 28 00:01:21,599 --> 00:01:23,339 sides of this equation. 29 00:01:23,340 --> 00:01:25,549 So what's the Laplace transform of 30 00:01:25,549 --> 00:01:26,609 the left-hand side? 31 00:01:26,609 --> 00:01:30,680 So the Laplace transform of the second derivative of y is 32 00:01:30,680 --> 00:01:34,530 just s squared, so now I'm taking the Laplace transform 33 00:01:34,530 --> 00:01:35,879 of just that. 34 00:01:35,879 --> 00:01:38,569 The Laplace transform of s squared times the Laplace 35 00:01:38,569 --> 00:01:45,109 transform of y minus-- lower the degree there once-- minus 36 00:01:45,109 --> 00:01:52,170 s times y of 0 minus y prime of 0. 37 00:01:52,170 --> 00:01:54,519 So clearly, I must have to give you some initial 38 00:01:54,519 --> 00:01:56,979 conditions in order to do this properly. 39 00:01:56,980 --> 00:02:05,490 And then plus 4 times the Laplace transform of y is 40 00:02:05,489 --> 00:02:10,978 equal to-- what's the Laplace transform of sine of t? 41 00:02:10,979 --> 00:02:12,690 That should be second nature by now. 42 00:02:12,689 --> 00:02:17,530 It's just 1 over s squared plus 1. 43 00:02:17,530 --> 00:02:20,159 And then we have minus the Laplace 44 00:02:20,159 --> 00:02:21,789 transform of this thing. 45 00:02:21,789 --> 00:02:25,389 And I'll do a little side note here to figure out the Laplace 46 00:02:25,389 --> 00:02:30,019 transform of this thing right here. 47 00:02:30,020 --> 00:02:32,630 And we know, I showed it to you a couple of videos ago, we 48 00:02:32,629 --> 00:02:39,319 showed that the Laplace transform-- actually I could 49 00:02:39,319 --> 00:02:42,370 just write it out here. 50 00:02:42,370 --> 00:02:45,289 This is going to be the same thing as the Laplace transform 51 00:02:45,289 --> 00:02:49,509 of sine of t, but we're going to have to multiply it by e to 52 00:02:49,509 --> 00:02:52,129 the minus-- if you remember that last formula-- e to the 53 00:02:52,129 --> 00:02:55,519 minus cs, where c is 2 pi. 54 00:02:55,520 --> 00:02:58,480 Actually, let me write that down. 55 00:02:58,479 --> 00:03:00,719 I decided to write it down, then I decided, oh, no, I 56 00:03:00,719 --> 00:03:02,080 don't want to do this. 57 00:03:02,080 --> 00:03:03,110 But let me write that. 58 00:03:03,110 --> 00:03:06,940 So the Laplace transform of the unit step function that 59 00:03:06,939 --> 00:03:12,949 goes up to c times some function shifted by c is equal 60 00:03:12,949 --> 00:03:20,239 to e to the minus cs times the Laplace transform of just the 61 00:03:20,240 --> 00:03:26,510 original function times the Laplace transform of f of t. 62 00:03:26,509 --> 00:03:28,599 So if we're taking the Laplace transform of this 63 00:03:28,599 --> 00:03:31,329 thing, our c is 2 pi. 64 00:03:31,330 --> 00:03:37,690 Our f of t is just sine of t, right? 65 00:03:37,689 --> 00:03:41,240 So then this is just going to be equal to-- if we just do 66 00:03:41,240 --> 00:03:45,629 this piece right here-- it's going to be equal to e to the 67 00:03:45,629 --> 00:03:53,479 minus cs-- our c is 2 pi-- e to the minus 2 pi s times the 68 00:03:53,479 --> 00:03:57,269 Laplace transform of f of t. f of t is just sine of t before 69 00:03:57,270 --> 00:03:58,120 we shifted. 70 00:03:58,120 --> 00:04:00,340 This is f of t minus 2 pi. 71 00:04:00,340 --> 00:04:03,090 So f of t is just going to be sine of t. 72 00:04:03,090 --> 00:04:07,340 So it's going to be times 1 over s squared plus 1. 73 00:04:07,340 --> 00:04:09,860 This is the Laplace transform of sine of t. 74 00:04:09,860 --> 00:04:14,000 So let's go back to where we had left off. 75 00:04:14,000 --> 00:04:16,490 So we've taken the Laplace transform of both sides of 76 00:04:16,490 --> 00:04:17,490 this equation. 77 00:04:17,490 --> 00:04:20,870 And clearly, I have some initial conditions here, so 78 00:04:20,870 --> 00:04:23,420 the problem must have given me some and I just forgot to 79 00:04:23,420 --> 00:04:25,290 write them down. 80 00:04:25,290 --> 00:04:28,189 So let's see, the initial conditions I'm given, and they 81 00:04:28,189 --> 00:04:31,149 are written kind of in the margin here, they tell us-- 82 00:04:31,149 --> 00:04:37,339 I'll do it in orange, they tell us that y of 0 is equal 83 00:04:37,339 --> 00:04:44,310 to 0, and y prime of 0 is equal to 0. 84 00:04:44,310 --> 00:04:46,089 That makes the math easy. 85 00:04:46,089 --> 00:04:49,109 That's 0 and that's 0. 86 00:04:49,110 --> 00:04:52,550 So let's see if I can simplify my equation. 87 00:04:52,550 --> 00:04:54,290 So the left-hand side, let's factor 88 00:04:54,290 --> 00:04:55,920 out the Laplace transform. 89 00:04:55,920 --> 00:04:59,439 So let's factor out this term and that term. 90 00:04:59,439 --> 00:05:06,759 So we get the Laplace transform of y times this plus 91 00:05:06,759 --> 00:05:13,740 this times s squared plus 4 is equal to the right-hand side. 92 00:05:13,740 --> 00:05:15,129 And what's the right-hand side? 93 00:05:15,129 --> 00:05:18,269 94 00:05:18,269 --> 00:05:19,569 We could simplify this. 95 00:05:19,569 --> 00:05:20,649 Well, I'll just write it out. 96 00:05:20,649 --> 00:05:22,169 I don't want to do too many steps at once. 97 00:05:22,170 --> 00:05:30,180 It's 1 over s squared plus 1 and then plus-- or minus 98 00:05:30,180 --> 00:05:32,420 actually, this is a minus-- minus the Laplace transfer of 99 00:05:32,420 --> 00:05:39,009 this thing, which was e to the minus 2 pi s over s 100 00:05:39,009 --> 00:05:41,719 squared plus 1. 101 00:05:41,720 --> 00:05:45,510 102 00:05:45,509 --> 00:05:48,189 So if we divide both sides of this equation by the s squared 103 00:05:48,189 --> 00:05:55,649 plus 4, then we get the Laplace transform of y is 104 00:05:55,649 --> 00:06:00,620 equal to-- and actually, I can just merge these two. 105 00:06:00,620 --> 00:06:01,810 They're the same denominator. 106 00:06:01,810 --> 00:06:04,699 So before I even divide by s squared plus 4, that 107 00:06:04,699 --> 00:06:07,089 right-hand side will look like this. 108 00:06:07,089 --> 00:06:11,409 It will look like with a denominator of s squared plus 109 00:06:11,410 --> 00:06:19,280 1 and you have a numerator of 1 minus e to the minus 2 pi s. 110 00:06:19,279 --> 00:06:21,149 And, of course, we're dividing both sides of this equation by 111 00:06:21,149 --> 00:06:23,259 s squared plus 4, so we're going to have to stick that s 112 00:06:23,259 --> 00:06:27,149 squared plus 4 over here. 113 00:06:27,149 --> 00:06:29,609 Now, we're at the hard part. 114 00:06:29,610 --> 00:06:31,910 In order to figure out why, we have to take the inverse 115 00:06:31,910 --> 00:06:34,140 Laplace transform of this thing. 116 00:06:34,139 --> 00:06:35,620 So how do we take the inverse Laplace 117 00:06:35,620 --> 00:06:36,720 transform of this thing? 118 00:06:36,720 --> 00:06:39,400 That's where the hard part is always, you know, it makes 119 00:06:39,399 --> 00:06:41,060 solving the differential equation's easy if you know 120 00:06:41,060 --> 00:06:45,213 the Laplace transforms. So it looks like we're going to have 121 00:06:45,213 --> 00:06:48,040 to do some partial fraction expansion. 122 00:06:48,040 --> 00:06:49,290 So let's see if we can do that. 123 00:06:49,290 --> 00:06:53,675 So we can rewrite this equation right here. 124 00:06:53,675 --> 00:06:56,829 125 00:06:56,829 --> 00:06:59,669 Actually, let's write it as this, because this'll kind of 126 00:06:59,670 --> 00:07:01,250 simplify our work. 127 00:07:01,250 --> 00:07:02,709 Let's factor this whole thing out. 128 00:07:02,709 --> 00:07:08,370 So we're going to write it as 1 minus e to the minus 2 pi s, 129 00:07:08,370 --> 00:07:13,340 all of that times-- I'll do it in orange-- all of that times 130 00:07:13,339 --> 00:07:21,939 1 over s squared plus 1 times s squared plus 4. 131 00:07:21,939 --> 00:07:24,410 Now, we need to do some partial fraction expansion to 132 00:07:24,410 --> 00:07:27,430 simplify this thing right here. 133 00:07:27,430 --> 00:07:29,259 We're going to do this on the side. 134 00:07:29,259 --> 00:07:32,389 Maybe I should do this over on the right here. 135 00:07:32,389 --> 00:07:38,300 This thing-- let me rewrite it-- 1 over s squared plus 1 136 00:07:38,300 --> 00:07:44,990 times s squared plus 4 should be able to be rewritten as two 137 00:07:44,990 --> 00:07:50,100 separate fractions, s squared plus 1 and s squared plus 4, 138 00:07:50,100 --> 00:07:52,680 with the numerators. 139 00:07:52,680 --> 00:07:56,069 This one would be As plus B. 140 00:07:56,069 --> 00:07:57,769 It's going to have to have degree 1, because 141 00:07:57,769 --> 00:07:58,599 this is degree 2. 142 00:07:58,600 --> 00:08:03,700 Here And then we'd have Cs plus D. 143 00:08:03,699 --> 00:08:11,360 And so when you add these two things up, you get As plus B 144 00:08:11,360 --> 00:08:23,000 times s squared plus 4 plus Cs plus D times s squared plus 1, 145 00:08:23,000 --> 00:08:27,870 all of that over the common denominator. 146 00:08:27,870 --> 00:08:31,329 We've seen this story before. 147 00:08:31,329 --> 00:08:33,649 We just have to do some algebra here. 148 00:08:33,649 --> 00:08:35,519 As you can tell, these differential equations 149 00:08:35,519 --> 00:08:37,189 problems, they require a lot of stamina. 150 00:08:37,190 --> 00:08:40,830 You kind of just have to say I will keep moving forward and 151 00:08:40,830 --> 00:08:43,700 do the algebra that I need to do in order to get the answer. 152 00:08:43,700 --> 00:08:45,950 And you kind of have to get excited about that notion that 153 00:08:45,950 --> 00:08:48,570 you have all this algebra to do. 154 00:08:48,570 --> 00:08:49,660 So let's figure it out. 155 00:08:49,659 --> 00:09:00,199 So this top can be simplified to As to the third plus Bs 156 00:09:00,200 --> 00:09:10,570 squared plus 4As plus 4B. 157 00:09:10,570 --> 00:09:19,330 And then this one, you end up with Cs to the third plus Ds 158 00:09:19,330 --> 00:09:27,850 squared plus Cs plus D. 159 00:09:27,850 --> 00:09:32,350 So when you add of these up together, you get-- and this 160 00:09:32,350 --> 00:09:35,409 is all the algebra that we have to do, for better, for 161 00:09:35,409 --> 00:09:43,509 worse-- A plus C over s to the third plus B plus D times s 162 00:09:43,509 --> 00:09:49,919 squared plus 4A plus C times s-- let's scroll over a little 163 00:09:49,919 --> 00:09:54,769 bit-- plus 4B plus D. 164 00:09:54,769 --> 00:09:59,730 And now we just have to say, OK, all of this is equal to 165 00:09:59,730 --> 00:10:00,710 this thing up here. 166 00:10:00,710 --> 00:10:01,769 This is the numerator. 167 00:10:01,769 --> 00:10:03,139 We just simplified the numerator. 168 00:10:03,139 --> 00:10:05,049 This is the numerator. 169 00:10:05,049 --> 00:10:06,839 That's the numerator right there. 170 00:10:06,840 --> 00:10:11,740 And all of this is going to be over your original s squared 171 00:10:11,740 --> 00:10:15,279 plus 1 times your s squared plus 4. 172 00:10:15,279 --> 00:10:18,149 And we established that this thing should be-- let me just 173 00:10:18,149 --> 00:10:23,610 write this-- that 1 over s squared plus 1 times s squared 174 00:10:23,610 --> 00:10:26,240 plus 4 should equal this thing. 175 00:10:26,240 --> 00:10:29,009 And then you just pattern match on the coefficients. 176 00:10:29,009 --> 00:10:33,419 This is all just intense partial fraction expansion. 177 00:10:33,419 --> 00:10:36,189 And you say, look, A plus C is the coefficient of the s cubed 178 00:10:36,190 --> 00:10:38,610 terms. I don't see any s cubed terms here. 179 00:10:38,610 --> 00:10:43,180 So A plus C must be equal to 0. 180 00:10:43,179 --> 00:10:46,069 And then you see, OK, B plus D is the coefficient of the s 181 00:10:46,070 --> 00:10:48,010 squared terms. I don't see any s squared terms there. 182 00:10:48,009 --> 00:10:53,669 So B plus D must be equal to 0. 183 00:10:53,669 --> 00:10:56,219 4A plus C, the coefficient of the s terms. I don't see any s 184 00:10:56,220 --> 00:10:57,070 terms over here. 185 00:10:57,070 --> 00:11:01,080 So 4A plus C must be equal to 0. 186 00:11:01,080 --> 00:11:02,490 And then we're almost done. 187 00:11:02,490 --> 00:11:05,490 4B plus D must be the constant terms. There is a constant 188 00:11:05,490 --> 00:11:06,529 term there. 189 00:11:06,529 --> 00:11:10,464 So 4B plus D is equal to 1. 190 00:11:10,465 --> 00:11:12,490 So let's see if we can do anything here. 191 00:11:12,490 --> 00:11:17,680 If we subtract this from that, we get minus 3A is equal to 0, 192 00:11:17,679 --> 00:11:19,159 or A is equal to 0. 193 00:11:19,159 --> 00:11:22,839 If A is equal to 0, then C is equals to 0. 194 00:11:22,840 --> 00:11:24,720 And let's see what we can get here. 195 00:11:24,720 --> 00:11:31,009 If we subtract this from that, we get minus 3B. 196 00:11:31,009 --> 00:11:32,019 The D's cancel out. 197 00:11:32,019 --> 00:11:36,289 It's equal to minus 1, or B is equal to 1/3. 198 00:11:36,289 --> 00:11:40,469 And then, of course, we have D is equal to minus B, if you 199 00:11:40,470 --> 00:11:44,550 subtract B from both sides. so D is equal to 1/3. 200 00:11:44,549 --> 00:11:46,959 So all of that work, and we actually have a 201 00:11:46,960 --> 00:11:48,400 pretty simple result. 202 00:11:48,399 --> 00:11:52,189 203 00:11:52,190 --> 00:12:00,660 Our equation, this thing here, can be rewritten as-- the A 204 00:12:00,659 --> 00:12:01,629 disappeared. 205 00:12:01,629 --> 00:12:06,129 It's 1/3 over s squared plus 1. 206 00:12:06,129 --> 00:12:10,490 B was the coefficient on the-- let me make it very clear. 207 00:12:10,490 --> 00:12:14,230 B was the coefficient on the-- or it was a term on top of the 208 00:12:14,230 --> 00:12:17,399 s squared plus 1, so that's why I'm using B there. 209 00:12:17,399 --> 00:12:22,730 And then D is minus B, so D is minus 1. 210 00:12:22,730 --> 00:12:23,509 So let me make sure I have that. 211 00:12:23,509 --> 00:12:29,059 B is 1/3 minus-- let me make sure I get that right. 212 00:12:29,059 --> 00:12:30,259 D is 1/3. 213 00:12:30,259 --> 00:12:36,830 So, sorry, B as in boy is 1/3, so D is minus 1/3. 214 00:12:36,830 --> 00:12:40,930 So B, there's a term on top of the s squared plus 1. 215 00:12:40,929 --> 00:12:45,559 And then you have minus D over the minus 1/3 over s 216 00:12:45,559 --> 00:12:50,509 squared plus 4. 217 00:12:50,509 --> 00:12:53,559 This takes a lot of stamina to record this video. 218 00:12:53,559 --> 00:12:56,369 I hope you appreciate it. 219 00:12:56,370 --> 00:12:59,090 OK, so let me rewrite everything, just so we can get 220 00:12:59,090 --> 00:13:01,450 back to the problem because when you take the partial 221 00:13:01,450 --> 00:13:05,740 fraction detour, you forget-- not even to speak of the 222 00:13:05,740 --> 00:13:07,620 problem, you forget what day it is. 223 00:13:07,620 --> 00:13:12,810 Let's see, so you get the Laplace transform of y is 224 00:13:12,809 --> 00:13:20,059 equal to 1 minus e to the minus 2 pi s times what that 225 00:13:20,059 --> 00:13:23,673 mess that we just solved for, times-- and I'll 226 00:13:23,673 --> 00:13:24,360 write it like this. 227 00:13:24,360 --> 00:13:34,440 1/3 times 1 over s squared plus 1 minus 1/3 times-- 228 00:13:34,440 --> 00:13:35,470 actually, let me write it this way. 229 00:13:35,470 --> 00:13:38,190 Because I have this s squared plus 4, so I really want to 230 00:13:38,190 --> 00:13:39,880 have a 2 there. 231 00:13:39,879 --> 00:13:41,679 So I want to have a 2 in the numerator, so you want to have 232 00:13:41,679 --> 00:13:44,824 a 2 over s squared plus 4. 233 00:13:44,825 --> 00:13:48,690 So if I put a 2 in the numerator there, I have to 234 00:13:48,690 --> 00:13:51,440 divide this by 2 as well. 235 00:13:51,440 --> 00:13:55,140 So let me change this to a 6. 236 00:13:55,139 --> 00:13:58,689 Minus 1/6 times 2 is minus 1/3. 237 00:13:58,690 --> 00:14:00,800 So I did that just so I get this in the form of the 238 00:14:00,799 --> 00:14:03,859 Laplace transform of sine of t. 239 00:14:03,860 --> 00:14:06,919 Now, let's see if there's anything that 240 00:14:06,919 --> 00:14:09,089 I can do from here. 241 00:14:09,090 --> 00:14:11,639 This is an epic problem. 242 00:14:11,639 --> 00:14:13,750 I'll be amazed if I don't make a careless 243 00:14:13,750 --> 00:14:16,320 mistake while I do this. 244 00:14:16,320 --> 00:14:19,170 So we can rewrite everything. 245 00:14:19,169 --> 00:14:21,279 Let's see if we can simplify this. 246 00:14:21,279 --> 00:14:23,879 And by simplifying it, I'm just going to make it longer. 247 00:14:23,879 --> 00:14:28,399 We can write the Laplace transform of y is equal to-- 248 00:14:28,399 --> 00:14:30,639 I'm just going to multiply the 1 out, and then I'm going to 249 00:14:30,639 --> 00:14:32,659 multiply the e to the minus 2 pi s out. 250 00:14:32,659 --> 00:14:39,539 So if you multiply the 1 out, you get 1/3 times 1 over s 251 00:14:39,539 --> 00:14:43,839 squared plus 1-- I'm just multiplying the 1 out-- minus 252 00:14:43,840 --> 00:14:50,060 1/6-- these are all the 1's times the 1-- times 2 over s 253 00:14:50,059 --> 00:14:52,019 squared plus 4. 254 00:14:52,019 --> 00:14:55,659 And then I'm going to multiply the minus e. 255 00:14:55,659 --> 00:14:58,669 Let me just switch colors, do the minus e. 256 00:14:58,669 --> 00:15:06,819 So then you get minus e to the minus 2 pi s over 3 times 1 257 00:15:06,820 --> 00:15:09,430 over s squared plus 1. 258 00:15:09,429 --> 00:15:11,859 And then the minus and the minus cancel out, so you get 259 00:15:11,860 --> 00:15:20,680 plus e to the minus 2 pi s over 6 times 2 over s 260 00:15:20,679 --> 00:15:22,679 squared plus 4. 261 00:15:22,679 --> 00:15:25,629 Now, taking the inverse Laplace transform of these 262 00:15:25,629 --> 00:15:26,799 things are pretty straightforward. 263 00:15:26,799 --> 00:15:27,579 So let's do that. 264 00:15:27,580 --> 00:15:31,000 Let's take the inverse Laplace transform of the whole thing. 265 00:15:31,000 --> 00:15:36,279 And we get y is equal to the inverse Laplace transform of 266 00:15:36,279 --> 00:15:44,389 this guy right here, is just 1/3 sine of t-- I don't have 267 00:15:44,389 --> 00:15:48,330 to write a parentheses there-- sine of t, and then this is 268 00:15:48,330 --> 00:15:56,590 minus 1/6 times-- this is the Laplace 269 00:15:56,590 --> 00:15:57,840 transform of sine of 2t. 270 00:15:57,840 --> 00:16:00,820 271 00:16:00,820 --> 00:16:02,730 That's that term right there. 272 00:16:02,730 --> 00:16:05,409 Now, these are almost the same, but we have this little 273 00:16:05,409 --> 00:16:06,699 pesky character over here. 274 00:16:06,700 --> 00:16:10,250 We have this e to the minus 2 pi s. 275 00:16:10,250 --> 00:16:12,485 And there, we just have to remind ourselves-- I'll write 276 00:16:12,485 --> 00:16:13,549 it here in the bottom. 277 00:16:13,549 --> 00:16:15,519 We just have to remind ourselves that the Laplace 278 00:16:15,519 --> 00:16:21,379 transform of the unit step function-- I'll put the pi 279 00:16:21,379 --> 00:16:28,470 there, just 2 pi times f of t minus 2 pi-- I should put as 280 00:16:28,470 --> 00:16:36,259 the step function of t-- is equal to e to the minus 2 pi s 281 00:16:36,259 --> 00:16:39,865 times the Laplace transform of just-- or let me just write it 282 00:16:39,865 --> 00:16:45,860 this way-- times the Laplace transform of f of t. 283 00:16:45,860 --> 00:16:50,039 So if we view f of t as just sine of t or sine of 2t, then 284 00:16:50,039 --> 00:16:52,289 we can kind of backwards pattern match. 285 00:16:52,289 --> 00:16:54,789 And we'll have to shift it and multiply it by 286 00:16:54,789 --> 00:16:58,500 the unit step function. 287 00:16:58,500 --> 00:17:00,049 So I want to make that clear. 288 00:17:00,049 --> 00:17:03,689 If you didn't have this guy here, the inverse Laplace 289 00:17:03,690 --> 00:17:06,009 transform of this guy would be the same thing as this guy. 290 00:17:06,009 --> 00:17:07,109 It'd just be sine of t. 291 00:17:07,109 --> 00:17:09,209 The inverse Laplace transform of this guy 292 00:17:09,210 --> 00:17:10,578 would be sine of 2t. 293 00:17:10,578 --> 00:17:13,450 But we have this pesky character here, which 294 00:17:13,450 --> 00:17:16,078 essentially, instead of having the inverse Laplace transform 295 00:17:16,078 --> 00:17:19,190 just being our f of t, it's going to be our f of t shifted 296 00:17:19,190 --> 00:17:23,509 by 2 pi times the unit step function, where 297 00:17:23,509 --> 00:17:26,049 it steps up at 2pi. 298 00:17:26,049 --> 00:17:34,339 So this is going to be minus 1/3 times the unit step 299 00:17:34,339 --> 00:17:41,009 function, where c is 2 pi of t times-- instead of sine of t-- 300 00:17:41,009 --> 00:17:46,150 sine of t minus 2pi. 301 00:17:46,150 --> 00:17:49,170 And then we're almost done. 302 00:17:49,170 --> 00:17:51,660 I'll do it in magenta to celebrate it. 303 00:17:51,660 --> 00:17:58,029 Plus this very last term, which is 1/6 times the unit 304 00:17:58,029 --> 00:18:03,549 step function 2 pi of t, the unit step function that steps 305 00:18:03,549 --> 00:18:13,779 up at 2 pi times sine of-- and we have to be careful here. 306 00:18:13,779 --> 00:18:16,990 Wherever we had a t before, we're going to replace it with 307 00:18:16,990 --> 00:18:18,750 a t minus 2 pi. 308 00:18:18,750 --> 00:18:20,849 So sine of, instead of 2t, is going to be 2 309 00:18:20,849 --> 00:18:25,269 times t minus 2 pi. 310 00:18:25,269 --> 00:18:26,539 And there you have it. 311 00:18:26,539 --> 00:18:31,649 We finally have solved our very hairy problem. 312 00:18:31,650 --> 00:18:34,600 We could take some time if we want to simplify 313 00:18:34,599 --> 00:18:35,250 this a little bit. 314 00:18:35,250 --> 00:18:36,779 In fact, we might as well. 315 00:18:36,779 --> 00:18:39,579 At the risk of making a careless mistake at the last 316 00:18:39,579 --> 00:18:42,689 moment, let me see if I can make any simplifications here. 317 00:18:42,690 --> 00:18:48,350 318 00:18:48,349 --> 00:18:52,469 Well, we could factor out this guy right here, but other than 319 00:18:52,470 --> 00:18:54,089 that, that seems about as simple as we can get. 320 00:18:54,089 --> 00:18:58,179 So this is our function of t that satisfies our otherwise 321 00:18:58,180 --> 00:19:01,330 simple-looking differential equation that we had up here. 322 00:19:01,329 --> 00:19:03,849 This looked fairly straightforward, but we got 323 00:19:03,849 --> 00:19:08,339 this big mess to actually satisfy that equation, given 324 00:19:08,339 --> 00:19:11,129 those initial conditions that we had initially.