1 00:00:00,000 --> 00:00:00,580 2 00:00:00,580 --> 00:00:03,879 It's been over a year since I last did a video with the 3 00:00:03,879 --> 00:00:05,419 differential equations playlist, and I thought I 4 00:00:05,419 --> 00:00:07,370 would start kicking up, making a couple of videos. 5 00:00:07,370 --> 00:00:10,839 And I think where I left, I said that I would do a 6 00:00:10,839 --> 00:00:15,339 non-homogenous linear equation using the Laplace Transform. 7 00:00:15,339 --> 00:00:18,039 So let's do one, as a bit of a warm-up, now that we've had 8 00:00:18,039 --> 00:00:20,049 a-- or at least I've had a one-year hiatus. 9 00:00:20,050 --> 00:00:21,660 Maybe you're watching these continuously, so you're 10 00:00:21,660 --> 00:00:24,039 probably more warmed up than I am. 11 00:00:24,039 --> 00:00:31,500 So if we have the equation the second derivative of y plus y 12 00:00:31,500 --> 00:00:36,820 is equal to sine of 2t. 13 00:00:36,820 --> 00:00:39,750 And we're given some initial conditions here. 14 00:00:39,750 --> 00:00:46,189 The initial conditions are y of 0 is equal to 2, and y 15 00:00:46,189 --> 00:00:50,879 prime of 0 is equal to 1. 16 00:00:50,880 --> 00:00:52,250 And where we left off-- and now you 17 00:00:52,250 --> 00:00:53,090 probably remember this. 18 00:00:53,090 --> 00:00:55,570 You probably recently watched the last video. 19 00:00:55,570 --> 00:00:56,939 To solve these, we just take the Laplace 20 00:00:56,939 --> 00:00:58,479 Transforms of all the sides. 21 00:00:58,479 --> 00:01:02,539 We solve for the Laplace Transform of the function. 22 00:01:02,539 --> 00:01:03,990 Then we take the inverse Laplace Transform. 23 00:01:03,990 --> 00:01:06,530 If that doesn't make sense, then let's just do it in this 24 00:01:06,530 --> 00:01:08,390 video, and hopefully the example 25 00:01:08,390 --> 00:01:11,189 will clarify all confusion. 26 00:01:11,189 --> 00:01:13,649 So in the last video-- it was either the last one or the 27 00:01:13,650 --> 00:01:18,460 previous one-- I showed you that the Laplace Transform of 28 00:01:18,459 --> 00:01:23,359 the second derivative of y is equal to s squared times the 29 00:01:23,359 --> 00:01:28,209 Laplace Transform of y-- and we keep lowering the degree on 30 00:01:28,209 --> 00:01:32,539 s-- so minus s times y of 0. 31 00:01:32,540 --> 00:01:34,530 You can kind of think of it as taking the derivative. 32 00:01:34,530 --> 00:01:35,510 This is an integral. 33 00:01:35,510 --> 00:01:38,260 It's not exactly the anti-derivative of this. 34 00:01:38,260 --> 00:01:40,329 But the Laplace Transform it is an integral. 35 00:01:40,329 --> 00:01:42,289 The Transform is an integral. 36 00:01:42,290 --> 00:01:47,530 So y of 0 is kind of one derivative away from that. 37 00:01:47,530 --> 00:01:52,400 And then minus y prime of 0. 38 00:01:52,400 --> 00:01:53,940 And then we could also rewrite this. 39 00:01:53,939 --> 00:01:56,269 And this is just a purely notational issue. 40 00:01:56,269 --> 00:01:59,299 I could write this, instead of writing the Laplace Transform 41 00:01:59,299 --> 00:02:02,599 of y all the time, I could write this as s squared times 42 00:02:02,599 --> 00:02:05,599 capital Y of s-- because this is going to be a function of 43 00:02:05,599 --> 00:02:12,699 s, not a function of y-- minus s times y of 0 minus 44 00:02:12,699 --> 00:02:14,669 y prime y of 0. 45 00:02:14,669 --> 00:02:16,509 These are going to be numbers, right? 46 00:02:16,509 --> 00:02:17,419 These aren't functions. 47 00:02:17,419 --> 00:02:22,500 These are the function evaluated at 0, or the 48 00:02:22,500 --> 00:02:24,189 derivative of the function evaluated at 0. 49 00:02:24,189 --> 00:02:25,689 And we know what these values are. 50 00:02:25,689 --> 00:02:31,909 y of 0, right here, is 2, and y prime of 0 is 1. 51 00:02:31,909 --> 00:02:33,340 It was given to us. 52 00:02:33,340 --> 00:02:35,740 So if we take the Laplace Transforms of both sides of 53 00:02:35,740 --> 00:02:37,939 this equation, first we're going to want to take the 54 00:02:37,939 --> 00:02:39,949 Laplace Transform of this term right there, which we've 55 00:02:39,949 --> 00:02:40,699 really just done. 56 00:02:40,699 --> 00:02:46,959 The Laplace Transform of the second derivative is s squared 57 00:02:46,960 --> 00:02:51,620 times the Laplace Transform of the function, which we write 58 00:02:51,620 --> 00:02:56,689 as capital Y of s, minus this, minus 2s-- they gave us that 59 00:02:56,689 --> 00:03:01,900 initial condition-- and then minus 1. 60 00:03:01,900 --> 00:03:02,659 Right? 61 00:03:02,659 --> 00:03:05,770 This term right here is just 1, so minus 1. 62 00:03:05,770 --> 00:03:07,560 So that's term right there. 63 00:03:07,560 --> 00:03:08,490 Then we want to take the Laplace 64 00:03:08,490 --> 00:03:11,379 Transform of y by itself. 65 00:03:11,379 --> 00:03:16,099 So this is just plus Y of s, right, the Laplace 66 00:03:16,099 --> 00:03:17,150 Transform of y. 67 00:03:17,150 --> 00:03:20,110 So I'll just rewrite Laplace Transform of y. 68 00:03:20,110 --> 00:03:22,500 I'm just rewriting it in this notation. 69 00:03:22,500 --> 00:03:23,199 Y of S. 70 00:03:23,199 --> 00:03:26,229 It's good to get used to either one. 71 00:03:26,229 --> 00:03:28,060 This is going to be equal to the Laplace 72 00:03:28,060 --> 00:03:31,469 Transform of sine of 2t. 73 00:03:31,469 --> 00:03:34,930 And I showed you in a video last year that we showed what 74 00:03:34,930 --> 00:03:37,740 the Laplace Transform of sine of at is, but I'll write it 75 00:03:37,740 --> 00:03:39,689 down here just so you remember it. 76 00:03:39,689 --> 00:03:49,169 Laplace Transform of the sine of at is equal to a over s 77 00:03:49,169 --> 00:03:52,324 squared plus a squared. 78 00:03:52,324 --> 00:03:56,039 79 00:03:56,039 --> 00:04:02,219 So the Laplace Transform of sine of 2t. 80 00:04:02,219 --> 00:04:03,379 Here, a is 2. 81 00:04:03,379 --> 00:04:07,139 This is going to be 2 over s squared plus 4. 82 00:04:07,139 --> 00:04:09,079 So if we take the Laplace Transform of both sides of 83 00:04:09,080 --> 00:04:13,630 this, the right-hand side is going to be 2 over s 84 00:04:13,629 --> 00:04:16,588 squared plus 4. 85 00:04:16,588 --> 00:04:19,389 Now what we can do is we can separate out all the Y of s 86 00:04:19,389 --> 00:04:23,240 terms. And so we can factor, well I guess we could say, 87 00:04:23,240 --> 00:04:25,269 factor out their coefficients, so that's a Y of s term, 88 00:04:25,269 --> 00:04:26,919 that's a Y of s term. 89 00:04:26,920 --> 00:04:30,170 And so we could write the left-hand side here as s 90 00:04:30,170 --> 00:04:34,610 squared-- that's that term-- plus 1-- the coefficient on 91 00:04:34,610 --> 00:04:39,685 that term-- s squared plus 1, times Y of s. 92 00:04:39,685 --> 00:04:41,600 Let me do it in green. 93 00:04:41,600 --> 00:04:47,450 So this is Y of s, and this Y of s, times Y of s, and then 94 00:04:47,449 --> 00:04:51,899 we have the non-Y of s terms. These two right here. 95 00:04:51,899 --> 00:04:58,769 So minus 2s, minus 1, is equal to 2 over s squared plus 4. 96 00:04:58,769 --> 00:05:01,490 We can add 2s plus 1 to both sides, to essentially move 97 00:05:01,490 --> 00:05:08,319 this to the right-hand side, and we're left with s squared 98 00:05:08,319 --> 00:05:20,389 plus 1, times Y of s, is equal to 2 over s squared plus 4, 99 00:05:20,389 --> 00:05:24,750 plus 2s, plus 1. 100 00:05:24,750 --> 00:05:28,009 Now we can divide both sides of this equation by s squared 101 00:05:28,009 --> 00:05:32,629 plus 1, and we get the Laplace Transform of Y. 102 00:05:32,629 --> 00:05:39,129 Y of s is equal to-- let me switch colors-- it's equal to 103 00:05:39,129 --> 00:05:45,879 2 over s squared plus 4 times this thing right here. 104 00:05:45,879 --> 00:05:48,889 I'm dividing both sides of this equation by this term 105 00:05:48,889 --> 00:05:49,539 right there. 106 00:05:49,540 --> 00:05:53,160 So times s squared plus 1-- it's in the denominator so I'm 107 00:05:53,160 --> 00:05:59,530 dividing by it-- plus 2s plus 1-- I have to divide both of 108 00:05:59,529 --> 00:06:03,809 those terms by the s squared plus 1-- divided by s squared 109 00:06:03,810 --> 00:06:09,290 plus 1, divided by s squared plus 1. 110 00:06:09,290 --> 00:06:11,830 Now, in order to be able to take the inverse Laplace 111 00:06:11,829 --> 00:06:15,349 Transform of this, I need to get it in some type of simple 112 00:06:15,350 --> 00:06:16,310 fraction form. 113 00:06:16,310 --> 00:06:18,069 These are actually easier to do, but this was one's a 114 00:06:18,069 --> 00:06:18,810 little bit difficult. 115 00:06:18,810 --> 00:06:21,790 I want to do some partial fraction decomposition to 116 00:06:21,790 --> 00:06:25,379 break this up into maybe simpler fractions. 117 00:06:25,379 --> 00:06:27,120 So what I want to do, I'm going to do a little bit of an 118 00:06:27,120 --> 00:06:28,100 aside here. 119 00:06:28,100 --> 00:06:31,890 And this really is the hardest part of these problems. The 120 00:06:31,889 --> 00:06:34,139 algebra, breaking this thing up. 121 00:06:34,139 --> 00:06:36,779 So I'm going to break this up. 122 00:06:36,779 --> 00:06:37,939 So let me write this this way. 123 00:06:37,939 --> 00:06:45,079 2 over s squared plus 4 times s squared plus 1. 124 00:06:45,079 --> 00:06:47,659 I'm going to break this up into two fractions. 125 00:06:47,660 --> 00:06:50,410 This is the partial fraction decomposition. 126 00:06:50,410 --> 00:06:52,660 One fraction is s squared plus 4. 127 00:06:52,660 --> 00:06:56,420 And the other fraction is s squared plus 1. 128 00:06:56,420 --> 00:07:02,110 And since both of these denominators are of degree 2, 129 00:07:02,110 --> 00:07:06,080 the numerators are going to be of degree 1. 130 00:07:06,079 --> 00:07:09,689 So they're going to be some-- let me write it this way-- 131 00:07:09,689 --> 00:07:12,629 this one will be As plus B. 132 00:07:12,629 --> 00:07:16,519 And then this one will be Cs plus D. 133 00:07:16,519 --> 00:07:17,969 This is just pure algebra. 134 00:07:17,970 --> 00:07:20,130 This is just partial fraction decomposition. 135 00:07:20,129 --> 00:07:22,319 I've made a couple of videos on it. 136 00:07:22,319 --> 00:07:25,759 And I'm saying that I'm assuming that this expression 137 00:07:25,759 --> 00:07:27,329 right here can be broken up into two 138 00:07:27,329 --> 00:07:28,789 expressions of this form. 139 00:07:28,790 --> 00:07:31,860 And I need to solve for A, B, C, and D. 140 00:07:31,860 --> 00:07:32,939 So let's see how we can do that. 141 00:07:32,939 --> 00:07:36,310 So if I were to start with these two and add them up, 142 00:07:36,310 --> 00:07:37,470 what do I get? 143 00:07:37,470 --> 00:07:40,820 I would have to multiply these times-- so my denominator, my 144 00:07:40,819 --> 00:07:46,040 common denominator, would be this thing again-- it would be 145 00:07:46,040 --> 00:07:53,530 s squared plus 4 times s squared plus 1. 146 00:07:53,529 --> 00:08:00,179 And now I'm going to have to multiply the As plus B times 147 00:08:00,180 --> 00:08:01,622 this s squared plus 1. 148 00:08:01,622 --> 00:08:05,039 149 00:08:05,040 --> 00:08:06,870 This, as it is right now, these two terms 150 00:08:06,870 --> 00:08:07,420 would cancel out. 151 00:08:07,420 --> 00:08:09,069 You'll just get this term, but I need to add it 152 00:08:09,069 --> 00:08:10,939 to this right here. 153 00:08:10,939 --> 00:08:24,969 So you get plus Cs plus D times s squared plus 4. 154 00:08:24,970 --> 00:08:27,660 And now let's see what we could do to match up the terms 155 00:08:27,660 --> 00:08:29,810 here with this number 2 right here. 156 00:08:29,810 --> 00:08:31,649 So let's multiply all of this out. 157 00:08:31,649 --> 00:08:39,779 So As times s squared is As to the 3rd. 158 00:08:39,779 --> 00:08:43,629 As times 1 is plus As. 159 00:08:43,629 --> 00:08:48,620 B times s squared, so plus Bs squared, and then you have B 160 00:08:48,620 --> 00:08:51,690 times 1 is plus B. 161 00:08:51,690 --> 00:08:54,525 And then you have Cs times s squared, that's Cs to the 3rd. 162 00:08:54,524 --> 00:08:58,129 163 00:08:58,129 --> 00:09:02,024 And then Cs times 4, so it's plus 4Cs. 164 00:09:02,024 --> 00:09:05,129 165 00:09:05,129 --> 00:09:08,240 These problems are tiring. 166 00:09:08,240 --> 00:09:11,560 And I also have a cold, so this is especially tiring, but 167 00:09:11,559 --> 00:09:13,409 I'll soldier forward. 168 00:09:13,409 --> 00:09:14,149 Where was I? 169 00:09:14,149 --> 00:09:16,199 So I multiplied the C's times each of these, now I have to 170 00:09:16,200 --> 00:09:17,040 multiply the D's. 171 00:09:17,039 --> 00:09:22,209 So plus Ds squared-- that's D times that one-- plus D times 172 00:09:22,210 --> 00:09:24,910 4, so plus 4D. 173 00:09:24,909 --> 00:09:25,949 So that's all of them. 174 00:09:25,950 --> 00:09:28,600 And I just wrote it this way so I have the common degree 175 00:09:28,600 --> 00:09:30,460 terms under each other. 176 00:09:30,460 --> 00:09:35,139 So if I were to add the entire numerator, I get-- and I'll 177 00:09:35,139 --> 00:09:42,100 just switch colors, somewhat arbitrarily-- I get A plus C 178 00:09:42,100 --> 00:09:46,580 times s to the 3rd plus-- let me write the s squared term 179 00:09:46,580 --> 00:09:51,509 next-- plus B plus D times s squared-- now I'll write this 180 00:09:51,509 --> 00:10:01,759 s term-- plus A plus 4C times s plus B plus 4D. 181 00:10:01,759 --> 00:10:02,870 This is just the numerator. 182 00:10:02,870 --> 00:10:04,710 This is when I just added these two things up. 183 00:10:04,710 --> 00:10:10,300 This whole thing up here simplifies to this. 184 00:10:10,299 --> 00:10:13,029 I don't know if the word simplify is appropriate. 185 00:10:13,029 --> 00:10:15,730 But it becomes this expression right here. 186 00:10:15,730 --> 00:10:16,960 And that's just the numerator. 187 00:10:16,960 --> 00:10:18,879 The denominator is still what we had written before. 188 00:10:18,879 --> 00:10:23,240 The denominator is still the s squared plus 4, times the s 189 00:10:23,240 --> 00:10:25,129 squared plus 1. 190 00:10:25,129 --> 00:10:28,720 Of course, I have to show that this is a fraction. 191 00:10:28,720 --> 00:10:31,100 And this is going to be equal to this thing over here. 192 00:10:31,100 --> 00:10:39,029 2 over s squared plus 4 times s squared plus 1. 193 00:10:39,029 --> 00:10:43,379 Now, why did I go through this whole mess right here? 194 00:10:43,379 --> 00:10:45,110 Well, the reason why I went through it is because we 195 00:10:45,110 --> 00:10:48,100 should be able to solve for A, B, C, and D. 196 00:10:48,100 --> 00:10:50,600 So let's see, A plus C. 197 00:10:50,600 --> 00:10:52,490 This is the coefficient on the s cubed term. 198 00:10:52,490 --> 00:10:54,850 Do we see any s cubed terms here? 199 00:10:54,850 --> 00:10:56,820 No, we see no s cubed terms here. 200 00:10:56,820 --> 00:11:02,180 So A plus C-- let me write this down-- A plus C must be 201 00:11:02,179 --> 00:11:03,899 equal to 0, because we see nothing here that 202 00:11:03,899 --> 00:11:05,990 has an s to the third. 203 00:11:05,990 --> 00:11:08,690 B plus D is a coefficient on the s squared term. 204 00:11:08,690 --> 00:11:10,720 Do we see any s squared terms here? 205 00:11:10,720 --> 00:11:13,120 No, so B plus D must be equal to 0. 206 00:11:13,120 --> 00:11:16,350 207 00:11:16,350 --> 00:11:23,259 A plus 4C are the coefficient of the s term. 208 00:11:23,259 --> 00:11:25,500 I see no s term over here. 209 00:11:25,500 --> 00:11:35,309 So A plus 4C must also be equal to 0. 210 00:11:35,309 --> 00:11:40,000 And then finally, we look at just the constant terms. And 211 00:11:40,000 --> 00:11:41,929 we do have a constant term on the left-hand 212 00:11:41,929 --> 00:11:42,889 side of this equation. 213 00:11:42,889 --> 00:11:44,039 We have 2. 214 00:11:44,039 --> 00:11:48,699 so B plus 4D-- I didn't want to make it that thick-- B plus 215 00:11:48,700 --> 00:11:53,295 4D must be equal to 2. 216 00:11:53,294 --> 00:11:58,379 217 00:11:58,379 --> 00:12:00,419 This just seems like these linear equations are pretty 218 00:12:00,419 --> 00:12:01,719 easy to solve for. 219 00:12:01,720 --> 00:12:06,149 Let's subtract this from this. 220 00:12:06,149 --> 00:12:08,779 So A-- or let me subtract the bottom one from the top one-- 221 00:12:08,779 --> 00:12:11,699 so A minus A, that's 0A. 222 00:12:11,700 --> 00:12:17,290 And then C minus 4C minus 3C is equal to 0. 223 00:12:17,289 --> 00:12:20,209 And so you get C is equal to 0. 224 00:12:20,210 --> 00:12:23,129 If C is equal to 0, A plus C is equals to 0, A must be 225 00:12:23,129 --> 00:12:25,809 equal to 0. 226 00:12:25,809 --> 00:12:27,459 And let's do the same thing here. 227 00:12:27,460 --> 00:12:29,889 Let's subtract this from that. 228 00:12:29,889 --> 00:12:35,610 So you get B minus B is 0, and then minus 3D-- that's just D 229 00:12:35,610 --> 00:12:39,899 minus 4D-- and then 0 minus 2 is equal to minus 2. 230 00:12:39,899 --> 00:12:48,079 And then you get D is equal to-- what do we get?-- D is 231 00:12:48,080 --> 00:12:50,080 equal to 2/3. 232 00:12:50,080 --> 00:12:53,190 233 00:12:53,190 --> 00:12:58,670 Minus 2 divided by minus 3 is 2/3, and then-- this isn't a 234 00:12:58,669 --> 00:13:01,099 minus here, I wrote that there later-- we said B plus D is 235 00:13:01,100 --> 00:13:01,899 equal to 0. 236 00:13:01,899 --> 00:13:04,370 So B must be the opposite of D, right? 237 00:13:04,370 --> 00:13:08,580 We could write B is equal to minus D, or B is 238 00:13:08,580 --> 00:13:11,150 equal to minus 2/3. 239 00:13:11,149 --> 00:13:14,509 Let's remember all of this and go back to our original 240 00:13:14,509 --> 00:13:17,509 problem, because we've kind of-- actually 241 00:13:17,509 --> 00:13:18,659 let me just be clear. 242 00:13:18,659 --> 00:13:26,110 We can rewrite 2 over s squared plus 4 times s 243 00:13:26,110 --> 00:13:28,149 squared plus 1. 244 00:13:28,149 --> 00:13:32,750 We can rewrite this as, well, A is 0, B is minus 2/3. 245 00:13:32,750 --> 00:13:39,710 So this is equal to minus 2/3 over s squared plus 4. 246 00:13:39,710 --> 00:13:42,450 And then C is 0, we figured that out. 247 00:13:42,450 --> 00:13:44,590 And then D is 2/3. 248 00:13:44,590 --> 00:13:51,530 So plus 2/3 over s squared plus 1. 249 00:13:51,529 --> 00:13:55,299 So all of that work that I just did, that was just to 250 00:13:55,299 --> 00:13:57,889 break up this piece right here. 251 00:13:57,889 --> 00:13:59,639 That was just to break up that piece right there. 252 00:13:59,639 --> 00:14:01,319 And then, of course, we have these other two pieces here 253 00:14:01,320 --> 00:14:02,820 that we can't forget about. 254 00:14:02,820 --> 00:14:05,550 So after all of this work, what do we have? 255 00:14:05,549 --> 00:14:06,469 And I'm going to make sure I don't make a 256 00:14:06,470 --> 00:14:08,389 careless mistake here. 257 00:14:08,389 --> 00:14:14,389 We get the Laplace Transform of Y-- as you can see, the 258 00:14:14,389 --> 00:14:20,669 algebra is the hardest part here-- is equal to this first 259 00:14:20,669 --> 00:14:23,929 term-- I'm just going back-- this first term, which I've 260 00:14:23,929 --> 00:14:27,879 now decomposed into this. 261 00:14:27,879 --> 00:14:32,850 So it's minus-- let me write it this way-- minus 1/3-- and 262 00:14:32,850 --> 00:14:34,820 I think you're going to see in a second why I'm writing this 263 00:14:34,820 --> 00:14:45,270 way-- minus 1/3 times 2 over s squared plus 4, and then plus 264 00:14:45,269 --> 00:14:51,600 2/3 times 1 over s squared plus 1. 265 00:14:51,600 --> 00:14:52,399 And you're probably saying, Sal, why are you 266 00:14:52,399 --> 00:14:53,470 writing it this way? 267 00:14:53,470 --> 00:14:56,250 Well you can already immediately see that this is 268 00:14:56,250 --> 00:14:59,029 the Laplace Transform of sine of 2t. 269 00:14:59,029 --> 00:15:01,529 This is the Laplace Transform of sine of t. 270 00:15:01,529 --> 00:15:03,750 So I wanted to write this 2 here, because this is 2, this 271 00:15:03,750 --> 00:15:04,370 is 2 squared. 272 00:15:04,370 --> 00:15:05,860 This is 1, this is 1 squared. 273 00:15:05,860 --> 00:15:07,759 So I wanted to write it in this form. 274 00:15:07,759 --> 00:15:08,909 This was just the first term. 275 00:15:08,909 --> 00:15:11,399 We have two more terms to worry about. 276 00:15:11,399 --> 00:15:12,769 I don't want to make a careless mistake. 277 00:15:12,769 --> 00:15:15,710 I have 2s over s squared plus 1. 278 00:15:15,710 --> 00:15:18,420 So let me write that down. 279 00:15:18,419 --> 00:15:31,349 So plus 2 times s over s squared plus 1, plus-- last 280 00:15:31,350 --> 00:15:34,730 one-- plus 1 over s squared plus 1. 281 00:15:34,730 --> 00:15:42,050 282 00:15:42,049 --> 00:15:45,669 Now we just take the inverse Laplace Transform of the whole 283 00:15:45,669 --> 00:15:49,329 thing, and then we'll know what Y is. 284 00:15:49,330 --> 00:15:52,810 285 00:15:52,809 --> 00:15:56,079 So, you know, just to remember the Laplace Transform. 286 00:15:56,080 --> 00:15:57,345 So this is going to be a little inverse. 287 00:15:57,345 --> 00:15:59,210 This is going to be sine of 2t. 288 00:15:59,210 --> 00:16:01,410 Let me just write, just so we have it here, so you know I'm 289 00:16:01,409 --> 00:16:03,089 not doing some type of voodoo. 290 00:16:03,090 --> 00:16:09,210 The Laplace Transform of sine of at is equal to a over s 291 00:16:09,210 --> 00:16:11,220 squared plus a squared. 292 00:16:11,220 --> 00:16:17,910 And the Laplace Transform of cosine of at is equal to s 293 00:16:17,909 --> 00:16:19,719 over s squared plus a squared. 294 00:16:19,720 --> 00:16:21,540 Let's just remember those two things when we take the 295 00:16:21,539 --> 00:16:25,679 inverse Laplace Transform of both sides of this equation. 296 00:16:25,679 --> 00:16:28,139 The inverse Laplace Transform of the Laplace Transform of y, 297 00:16:28,139 --> 00:16:30,539 well that's just y. 298 00:16:30,539 --> 00:16:35,009 y-- maybe I'll write it as a function of t-- is equal to-- 299 00:16:35,009 --> 00:16:38,210 well this is the Laplace Transform of sine of 2t. 300 00:16:38,210 --> 00:16:40,730 You can just do some pattern matching right here. 301 00:16:40,730 --> 00:16:43,450 If a is equal to 2, then this would be the Laplace Transform 302 00:16:43,450 --> 00:16:44,520 of sine of 2t. 303 00:16:44,519 --> 00:16:55,799 So it's minus 1/3 times sine of 2t plus 2/3 times-- this is 304 00:16:55,799 --> 00:16:57,309 the Laplace Transform of sine of t. 305 00:16:57,309 --> 00:17:00,099 If you just make a is equal to 1, sine of t's Laplace 306 00:17:00,100 --> 00:17:02,440 Transform is 1 over s squared plus 1. 307 00:17:02,440 --> 00:17:08,140 So plus 2/3 times the sine of t-- let me do the next one in 308 00:17:08,140 --> 00:17:12,430 blue, just because it was already written in blue-- plus 309 00:17:12,430 --> 00:17:15,779 2 times-- this is the Laplace Transform of cosine of t. 310 00:17:15,779 --> 00:17:19,078 If you make a is equal to 1, then the cosine t Laplace 311 00:17:19,078 --> 00:17:21,139 Transform is s over s squared plus 1. 312 00:17:21,140 --> 00:17:27,980 So 2 times cosine of t-- and then one last term-- plus-- 313 00:17:27,980 --> 00:17:30,150 this is just like this one over here, this is just the 314 00:17:30,150 --> 00:17:35,470 Laplace Transform of sine of t-- plus sine of t. 315 00:17:35,470 --> 00:17:36,569 And we're almost done. 316 00:17:36,569 --> 00:17:38,269 We're essentially done, but there's a little bit more 317 00:17:38,269 --> 00:17:40,289 simplification we can do. 318 00:17:40,289 --> 00:17:43,819 I have 2/3 times the sign of t here, and then I have another 319 00:17:43,819 --> 00:17:48,809 1 sine of t here, so I can add the 2/3 to the 1. 320 00:17:48,809 --> 00:17:51,609 What's 2/3 plus 1, or 3/3? 321 00:17:51,609 --> 00:17:52,609 It's 5/3. 322 00:17:52,609 --> 00:18:01,269 So I can write y of t is equal to minus 1/3 sine of 2t plus-- 323 00:18:01,269 --> 00:18:03,430 these two terms I'm just going to add up-- 324 00:18:03,430 --> 00:18:07,720 plus 5/3 sine of t. 325 00:18:07,720 --> 00:18:13,350 And then I have this last term here, plus 2cosine of t. 326 00:18:13,349 --> 00:18:15,949 So this was a hairy problem, a lot of work. 327 00:18:15,950 --> 00:18:18,340 And we saw that the hardest part really was just the 328 00:18:18,339 --> 00:18:21,699 partial fraction decomposition that we did up here, not 329 00:18:21,700 --> 00:18:23,110 making any careless mistakes. 330 00:18:23,109 --> 00:18:25,899 But at the end, we got a pretty neat answer that's not 331 00:18:25,900 --> 00:18:33,060 too complicated, that satisfies this non-homogenous 332 00:18:33,059 --> 00:18:34,419 differential equation. 333 00:18:34,420 --> 00:18:36,539 We were able to incorporate the boundary 334 00:18:36,539 --> 00:18:38,450 conditions as we did it. 335 00:18:38,450 --> 00:18:43,184 Anyway, hopefully you found that vaguely satisfying. 336 00:18:43,184 --> 00:18:45,769 This is a good warm-up after a year of no 337 00:18:45,769 --> 00:18:47,019 differential equations. 338 00:18:47,019 --> 00:18:47,826