1 00:00:00,090 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,150 to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,310 at ocw.mit.edu. 8 00:00:21,530 --> 00:00:24,580 PROFESSOR: Well, OK, Professor Frey 9 00:00:24,580 --> 00:00:27,840 invited me to give the two lectures this week 10 00:00:27,840 --> 00:00:35,610 on first order equations, like that one, first order dy dt. 11 00:00:35,610 --> 00:00:39,560 And the lectures next week will be on second order equation. 12 00:00:39,560 --> 00:00:45,380 So we're looking for, you could say, formulas for the solution. 13 00:00:45,380 --> 00:00:48,460 We'll get as far as we can with formulas, then 14 00:00:48,460 --> 00:00:49,500 numerical methods. 15 00:00:49,500 --> 00:00:54,520 Graphical methods take over in more complicated problems. 16 00:00:54,520 --> 00:00:57,560 This is a model problem. 17 00:00:57,560 --> 00:00:58,140 It's linear. 18 00:01:01,320 --> 00:01:05,010 I chose it to have constant coefficient a, 19 00:01:05,010 --> 00:01:08,760 and let me check the units. 20 00:01:08,760 --> 00:01:11,990 Always good to see the units in a problem. 21 00:01:11,990 --> 00:01:17,780 So let me think of this y, as the money in a bank, 22 00:01:17,780 --> 00:01:26,930 or bank balance, so y as in dollars, and t, time, in years. 23 00:01:26,930 --> 00:01:35,070 So we're looking at the ups and downs of bank balance y. 24 00:01:35,070 --> 00:01:43,300 The rate of change, so the units then are dollars per year. 25 00:01:43,300 --> 00:01:47,010 So every term in the equation has to have the right units. 26 00:01:47,010 --> 00:01:51,600 So y is in dollars, so the interest rate a 27 00:01:51,600 --> 00:01:56,750 is percent per year, say 6% a year. 28 00:01:56,750 --> 00:02:02,400 So a could be 6%-- that's dimensionless-- per year, 29 00:02:02,400 --> 00:02:10,430 or half a percent per month if we change. 30 00:02:10,430 --> 00:02:12,830 So if we change units, the constant a 31 00:02:12,830 --> 00:02:14,580 would change from 6 to a half. 32 00:02:14,580 --> 00:02:17,170 But let's stay with 6. 33 00:02:17,170 --> 00:02:24,240 And then q of t represents deposits and withdrawals, 34 00:02:24,240 --> 00:02:27,760 so that's in dollars per year again. 35 00:02:27,760 --> 00:02:28,890 Has to be. 36 00:02:28,890 --> 00:02:34,050 So that's continuous. 37 00:02:34,050 --> 00:02:38,330 We think of the deposits and the interest 38 00:02:38,330 --> 00:02:44,840 as being computed continuously as time goes forward. 39 00:02:44,840 --> 00:02:48,190 So if that's a constant-- and I'll take that case first, 40 00:02:48,190 --> 00:02:51,700 q equal 1-- that would mean that we're putting in, 41 00:02:51,700 --> 00:02:57,330 depositing $1 per year, continuously through the year. 42 00:02:57,330 --> 00:03:00,750 So that's the model that comes from a differential equation. 43 00:03:00,750 --> 00:03:05,980 A difference equation would give us finite time steps. 44 00:03:05,980 --> 00:03:10,050 So I'm looking for the solution. 45 00:03:10,050 --> 00:03:15,210 And with constant coefficients, linear, we're 46 00:03:15,210 --> 00:03:18,610 going to get a formula for the solution. 47 00:03:18,610 --> 00:03:21,750 I could actually deal with variable interest 48 00:03:21,750 --> 00:03:26,120 rate for this one first order equation, 49 00:03:26,120 --> 00:03:27,750 but the formula becomes messy. 50 00:03:27,750 --> 00:03:29,760 But you can still do it. 51 00:03:29,760 --> 00:03:32,210 After that point, for a second order equations 52 00:03:32,210 --> 00:03:36,210 like oscillation, or for a system 53 00:03:36,210 --> 00:03:40,480 of several equations coupled together, 54 00:03:40,480 --> 00:03:44,690 constant coefficients is where you can get formulas. 55 00:03:44,690 --> 00:03:47,320 So let's go with that case. 56 00:03:47,320 --> 00:03:49,400 So how to solve that equation? 57 00:03:52,030 --> 00:03:57,810 Let me take first of all, a constant, constant source. 58 00:03:57,810 --> 00:04:01,260 So I think of q as the source term. 59 00:04:01,260 --> 00:04:08,080 To get one nice formula, let me take this example, ay plus 1, 60 00:04:08,080 --> 00:04:09,361 let's say. 61 00:04:09,361 --> 00:04:12,240 How do you find y of t to solve that? 62 00:04:12,240 --> 00:04:15,960 And you start with some initial condition y of 0. 63 00:04:15,960 --> 00:04:21,130 That's the opening deposit that you make at time 0. 64 00:04:21,130 --> 00:04:23,410 How to solve that equation? 65 00:04:23,410 --> 00:04:32,380 Well, we're looking for a solution. 66 00:04:32,380 --> 00:04:36,240 And solutions to linear equations have two parts. 67 00:04:36,240 --> 00:04:40,290 So the same will happen in linear algebra. 68 00:04:40,290 --> 00:04:43,660 One part is a solution to that equation, 69 00:04:43,660 --> 00:04:47,550 so we're just looking for one, any one, 70 00:04:47,550 --> 00:04:49,940 and we'll call it a particular solution. 71 00:04:49,940 --> 00:04:56,350 And the associated null equation, dy dt equal ay. 72 00:04:59,240 --> 00:05:04,350 So this is an equation with q equals 0. 73 00:05:04,350 --> 00:05:07,260 That's why it's called null. 74 00:05:07,260 --> 00:05:10,630 And it's also called homogeneous. 75 00:05:10,630 --> 00:05:14,390 So more textbooks use that long word homogeneous, 76 00:05:14,390 --> 00:05:18,030 but I use the word null because it's shorter 77 00:05:18,030 --> 00:05:23,410 and because it's the same word in linear algebra. 78 00:05:26,970 --> 00:05:30,620 So let me call yn the null solution, 79 00:05:30,620 --> 00:05:32,530 the general null solution. 80 00:05:32,530 --> 00:05:37,270 And y, I'm looking here for a particular solution yp, 81 00:05:37,270 --> 00:05:41,710 and I'm going to-- here's the key for linear equations. 82 00:05:41,710 --> 00:05:47,665 Let me take that off and focus on those two equations. 83 00:05:50,300 --> 00:05:54,390 How does solving the null equation, which is easy to do, 84 00:05:54,390 --> 00:05:56,080 help me? 85 00:05:56,080 --> 00:06:02,870 Why can I, as I plan to do, add in yn to yp? 86 00:06:02,870 --> 00:06:04,910 I just add the two equations. 87 00:06:04,910 --> 00:06:08,490 Can I just add those two equations? 88 00:06:08,490 --> 00:06:17,090 I get the derivative of yp plus yn on the left side. 89 00:06:17,090 --> 00:06:22,110 And I have a times yp plus yn. 90 00:06:22,110 --> 00:06:27,190 And that is a critical moment there when we use linearity. 91 00:06:27,190 --> 00:06:33,410 I had a yp a yn, and I could put them together. 92 00:06:33,410 --> 00:06:38,070 If it was y squared, yp squared and yn squared 93 00:06:38,070 --> 00:06:42,580 would not be the same as yp plus yn squared. 94 00:06:42,580 --> 00:06:48,490 It's the linearity that comes, and then I add the 1. 95 00:06:48,490 --> 00:06:51,100 So what do I see from this? 96 00:06:51,100 --> 00:06:56,840 I see that yp plus yn also solves my equation. 97 00:06:56,840 --> 00:07:04,600 So the whole family of solutions is 1 yp plus any yn. 98 00:07:04,600 --> 00:07:06,480 And why do I say any yn? 99 00:07:06,480 --> 00:07:11,470 Because when I find one, I find more. 100 00:07:11,470 --> 00:07:14,930 The solutions to this equation are 101 00:07:14,930 --> 00:07:20,940 yn could be e to the at, because the derivative of e 102 00:07:20,940 --> 00:07:23,870 to the at does bring down a factor a. 103 00:07:23,870 --> 00:07:29,940 But you see, I've left space for any multiple of e to the at. 104 00:07:29,940 --> 00:07:33,580 This is where that long word homogeneous comes from. 105 00:07:33,580 --> 00:07:37,650 It's homogeneous means I can multiply by any constant, 106 00:07:37,650 --> 00:07:39,630 and I still solve the equation. 107 00:07:39,630 --> 00:07:43,710 And of course, the key again is linear. 108 00:07:43,710 --> 00:07:49,420 So now I have-- well, you could say I've done half the job. 109 00:07:49,420 --> 00:07:53,210 I've found yn, the general yn. 110 00:07:53,210 --> 00:07:57,410 And now I just have to find one yp, 111 00:07:57,410 --> 00:08:01,150 one solution to the equation. 112 00:08:01,150 --> 00:08:08,100 And with this source term, a constant, 113 00:08:08,100 --> 00:08:11,690 there's a nice way to find that solution. 114 00:08:11,690 --> 00:08:14,210 Look for a constant solution. 115 00:08:14,210 --> 00:08:16,970 So certain right hand sides, and those 116 00:08:16,970 --> 00:08:19,450 are the like the special functions 117 00:08:19,450 --> 00:08:23,080 for the special source terms for differential equation, 118 00:08:23,080 --> 00:08:24,780 certain right hand sides-- and I'm just 119 00:08:24,780 --> 00:08:27,750 going to go down a list of them today. 120 00:08:27,750 --> 00:08:29,610 The next one on the list-- can I tell you 121 00:08:29,610 --> 00:08:32,320 what the next one on the list will be? 122 00:08:32,320 --> 00:08:35,659 y prime equal ay. 123 00:08:35,659 --> 00:08:41,070 I use prime for-- well, I'll write dy dt, 124 00:08:41,070 --> 00:08:43,010 but often I'll write y prime. 125 00:08:43,010 --> 00:08:46,230 dy dt equal ay plus an exponential. 126 00:08:49,200 --> 00:08:51,440 That'll be number two. 127 00:08:51,440 --> 00:08:55,290 So I'm just preparing the way for number two. 128 00:08:55,290 --> 00:08:59,240 Well, actually number one, this example 129 00:08:59,240 --> 00:09:02,580 is the same as that exponential example with exponent 130 00:09:02,580 --> 00:09:06,230 s equal 0, right? 131 00:09:06,230 --> 00:09:09,320 If s is 0, then I have a constant. 132 00:09:09,320 --> 00:09:11,840 So this is a special case of that one. 133 00:09:11,840 --> 00:09:15,080 This is the most important source term 134 00:09:15,080 --> 00:09:18,450 in the whole subject. 135 00:09:18,450 --> 00:09:23,810 But here we go with a constant 1. 136 00:09:23,810 --> 00:09:25,480 So we've got yn. 137 00:09:25,480 --> 00:09:26,380 And what's yp? 138 00:09:31,870 --> 00:09:33,550 I just looked to see. 139 00:09:33,550 --> 00:09:35,500 Can I think of one? 140 00:09:35,500 --> 00:09:38,630 And with these special functions, 141 00:09:38,630 --> 00:09:43,200 you can often find a solution of the same form 142 00:09:43,200 --> 00:09:45,810 as the source term. 143 00:09:45,810 --> 00:09:48,830 And in this case, that means a constant. 144 00:09:48,830 --> 00:09:52,120 So if yp is a constant, this will be 0. 145 00:09:52,120 --> 00:09:54,100 So I just want to pick the constant that 146 00:09:54,100 --> 00:09:55,770 makes this thing 0. 147 00:09:55,770 --> 00:10:03,160 And of course, their right hand side is 0 when yp is minus 1 148 00:10:03,160 --> 00:10:04,780 over a. 149 00:10:04,780 --> 00:10:06,920 So I've got it. 150 00:10:06,920 --> 00:10:09,590 We've solved that equation, except we 151 00:10:09,590 --> 00:10:12,080 didn't match the initial condition yet. 152 00:10:12,080 --> 00:10:14,570 Let me if you take that final step. 153 00:10:14,570 --> 00:10:22,220 So the general y is any multiple, any null solution, 154 00:10:22,220 --> 00:10:27,260 plus any one particular solution, that one. 155 00:10:27,260 --> 00:10:33,650 And we want to match it to y of 0 at t equals 0. 156 00:10:33,650 --> 00:10:36,600 So I want to take that solution. 157 00:10:36,600 --> 00:10:38,530 I want to find that constant, here. 158 00:10:38,530 --> 00:10:42,190 That's the only remaining step is find that constant. 159 00:10:42,190 --> 00:10:44,430 You've done it in the homework. 160 00:10:44,430 --> 00:10:51,600 So at t equals 0, y of 0 is-- at t 161 00:10:51,600 --> 00:10:55,880 equals 0, this is C. This is the minus 1 over a. 162 00:10:55,880 --> 00:10:59,210 So I learn what the C has to be. 163 00:10:59,210 --> 00:11:00,870 And that's the final step. 164 00:11:00,870 --> 00:11:04,490 C is bring the 1 over a onto that side, 165 00:11:04,490 --> 00:11:18,130 so C will be y of 0 e to the at minus 1 over a e to at. 166 00:11:18,130 --> 00:11:22,310 And here we had a minus 1 over a. 167 00:11:22,310 --> 00:11:28,590 Well, it'll be plus 1 over a e to the at. 168 00:11:28,590 --> 00:11:35,190 So now I've just put in the C, y of 0 plus 1 over a. y of 0 169 00:11:35,190 --> 00:11:37,880 plus 1 over a has gone in for C. And now I 170 00:11:37,880 --> 00:11:40,140 have to subtract this 1 over a. 171 00:11:40,140 --> 00:11:43,670 Here, I see a 1 over a, so I can do it neatly. 172 00:11:48,990 --> 00:11:50,310 Got a solution. 173 00:11:50,310 --> 00:11:51,790 We can check it, of course. 174 00:11:51,790 --> 00:11:58,900 At t equals 0, this disappears, and this is y of 0. 175 00:11:58,900 --> 00:12:00,380 And it has the form. 176 00:12:00,380 --> 00:12:06,030 It's a multiple of e to the at and a particular solution. 177 00:12:06,030 --> 00:12:06,960 So that's a good one. 178 00:12:11,190 --> 00:12:16,810 Notice that to get the initial condition right, 179 00:12:16,810 --> 00:12:20,440 I couldn't take C to be y of 0 to get the initial condition 180 00:12:20,440 --> 00:12:21,240 right. 181 00:12:21,240 --> 00:12:22,750 To get the initial condition right, 182 00:12:22,750 --> 00:12:28,580 I had to get that, this minus 1 over a in there. 183 00:12:28,580 --> 00:12:29,400 Good for that one? 184 00:12:32,160 --> 00:12:36,230 Let me move to the next one, exponentials. 185 00:12:40,220 --> 00:12:48,220 So again, we know that the null equation with no source 186 00:12:48,220 --> 00:12:52,240 has this solution e to the at. 187 00:12:52,240 --> 00:12:57,410 And I'm going to suppose that the a in e to the 188 00:12:57,410 --> 00:13:02,794 at in the null solution is different from the s 189 00:13:02,794 --> 00:13:07,070 in the source function, which will come up 190 00:13:07,070 --> 00:13:09,200 in the particular solution. 191 00:13:09,200 --> 00:13:11,910 So you're going to see either the st 192 00:13:11,910 --> 00:13:14,530 in the particular solution and an e to the 193 00:13:14,530 --> 00:13:17,230 at in the null solution. 194 00:13:17,230 --> 00:13:24,260 And in the case when s equals a, that's called resonance, 195 00:13:24,260 --> 00:13:28,720 the two exponents are the same, and the formula changes 196 00:13:28,720 --> 00:13:30,700 a little. 197 00:13:30,700 --> 00:13:33,850 Let's leave that case for later. 198 00:13:33,850 --> 00:13:36,590 How do I solve this? 199 00:13:36,590 --> 00:13:38,690 I'm looking for a particular solution 200 00:13:38,690 --> 00:13:42,140 because I know the null solutions. 201 00:13:42,140 --> 00:13:44,690 How am I going to get a particular solution 202 00:13:44,690 --> 00:13:46,170 of this equation? 203 00:13:46,170 --> 00:13:50,970 Fundamental observation, the key point 204 00:13:50,970 --> 00:13:56,140 is it's going to be a multiple of e to the st. 205 00:13:56,140 --> 00:14:02,470 If an exponential goes in, then that will be an exponential. 206 00:14:02,470 --> 00:14:04,600 Its derivative will be an exponential. 207 00:14:04,600 --> 00:14:08,920 I'll have e to the st's everywhere. 208 00:14:08,920 --> 00:14:12,910 And I can get the number right. 209 00:14:12,910 --> 00:14:15,950 So I'm looking for y try. 210 00:14:15,950 --> 00:14:24,180 So I'll put try, knowing it's going to work, as some number 211 00:14:24,180 --> 00:14:31,310 times e to the st. So this would be 212 00:14:31,310 --> 00:14:33,580 like the exponential response. 213 00:14:33,580 --> 00:14:36,370 Response, do you know that word response? 214 00:14:36,370 --> 00:14:39,820 So response is the solution. 215 00:14:39,820 --> 00:14:44,510 The input is q, and the response is Y. 216 00:14:44,510 --> 00:14:47,036 And here, the input is e to the st, 217 00:14:47,036 --> 00:14:53,550 and the response is a multiple of e to the st. So plug it in. 218 00:14:53,550 --> 00:14:58,940 The timed derivative will be Y. Taking 219 00:14:58,940 --> 00:15:04,000 the derivative will bring down a 1. e to the st equals aY. 220 00:15:04,000 --> 00:15:12,090 A aY e to the st plus 1 e to the st. Just what we hoped. 221 00:15:15,350 --> 00:15:18,980 The beauty of exponentials is that when 222 00:15:18,980 --> 00:15:21,710 you take their derivatives, you just have more exponential. 223 00:15:21,710 --> 00:15:23,770 That's the key thing. 224 00:15:23,770 --> 00:15:26,580 That's why exponential is the most important function 225 00:15:26,580 --> 00:15:30,610 in this course, absolutely the most important function. 226 00:15:30,610 --> 00:15:33,360 So it happened here. 227 00:15:33,360 --> 00:15:36,120 I can cancel e to the st, because every term 228 00:15:36,120 --> 00:15:37,430 has one of them. 229 00:15:37,430 --> 00:15:41,070 So I'm seeing that-- what am I getting for Y? 230 00:15:41,070 --> 00:15:44,080 Getting a very important number for Y. So 231 00:15:44,080 --> 00:15:48,050 I bring aY onto this side with sY. 232 00:15:48,050 --> 00:15:49,710 On this side I just have a 1. 233 00:15:52,540 --> 00:15:55,560 Maybe it's worth putting on its own board. 234 00:15:55,560 --> 00:16:06,080 Y is, so Ys aY comes with a minus, and the 1, 1 over-- 235 00:16:06,080 --> 00:16:08,760 so Y was multiplied by s minus a. 236 00:16:12,130 --> 00:16:17,610 That's the right quantity to get a particular solution. 237 00:16:17,610 --> 00:16:21,490 And that 1 over s minus a, you see 238 00:16:21,490 --> 00:16:24,490 why I wanted s to be different from a. 239 00:16:24,490 --> 00:16:28,630 I If s equaled a in that case, in that possibility 240 00:16:28,630 --> 00:16:32,620 of resonance when the two exponents are the same, 241 00:16:32,620 --> 00:16:35,630 we would have 1 over 0, and we'd have to look somewhere else. 242 00:16:39,690 --> 00:16:43,160 The name for that-- this has to have a name because it shows up 243 00:16:43,160 --> 00:16:43,890 all the time. 244 00:16:43,890 --> 00:16:46,220 The exponential response function, 245 00:16:46,220 --> 00:16:47,660 you could call it that. 246 00:16:47,660 --> 00:16:49,995 Most people would call it the transfer function. 247 00:16:57,210 --> 00:17:00,130 So any constant coefficient linear equation's 248 00:17:00,130 --> 00:17:03,610 going to have a transfer function, easy to find. 249 00:17:03,610 --> 00:17:06,520 Everything easy, that's what I'm emphasizing, here. 250 00:17:06,520 --> 00:17:08,960 Everything's straightforward. 251 00:17:08,960 --> 00:17:12,540 That transfer function tells you what 252 00:17:12,540 --> 00:17:14,780 multiplies the exponential. 253 00:17:14,780 --> 00:17:18,200 So the source was here. 254 00:17:20,720 --> 00:17:28,640 And the response is here, the response factor, you could say, 255 00:17:28,640 --> 00:17:29,800 the transfer function. 256 00:17:29,800 --> 00:17:32,510 Multiply by 1 over s minus a. 257 00:17:32,510 --> 00:17:37,600 So if s is close to a, if the input is almost 258 00:17:37,600 --> 00:17:45,960 at the same exponent as the natural, as the null solution, 259 00:17:45,960 --> 00:17:50,290 then we're going to get a big response. 260 00:17:50,290 --> 00:17:52,180 So that's good. 261 00:17:52,180 --> 00:17:55,490 For a constant coefficient problem second order, 262 00:17:55,490 --> 00:17:58,560 other problems we can find that response function. 263 00:17:58,560 --> 00:18:00,520 It's the key function. 264 00:18:00,520 --> 00:18:05,640 It's the function if we have, or if we 265 00:18:05,640 --> 00:18:09,580 were to look at Laplace transforms, that 266 00:18:09,580 --> 00:18:11,540 would be the key. 267 00:18:11,540 --> 00:18:13,690 When you take Laplace transforms, 268 00:18:13,690 --> 00:18:16,800 the transfer function shows up. 269 00:18:16,800 --> 00:18:19,150 Then when you take inverse Laplace transforms, 270 00:18:19,150 --> 00:18:23,245 you have to find what function has that Laplace transform. 271 00:18:26,670 --> 00:18:30,660 So did we get the-- we got the final answer then. 272 00:18:30,660 --> 00:18:36,980 Let me put it here. y is e to the st times this factor. 273 00:18:36,980 --> 00:18:39,255 So I divide by s minus a. 274 00:18:39,255 --> 00:18:41,210 A nice solution. 275 00:18:46,160 --> 00:18:49,220 Let me also anticipate something more. 276 00:18:51,960 --> 00:18:59,510 An important case for e to the st is e to the i omega t. 277 00:18:59,510 --> 00:19:03,580 e to the st, we think about as exponential growth, 278 00:19:03,580 --> 00:19:05,380 exponential decay. 279 00:19:05,380 --> 00:19:10,200 But that's for positive s and negative s. 280 00:19:10,200 --> 00:19:16,570 And all important in applications is oscillation. 281 00:19:16,570 --> 00:19:25,740 So coming, let me say, coming is either late today 282 00:19:25,740 --> 00:19:31,980 or early Wednesday will be s equal i omega, 283 00:19:31,980 --> 00:19:41,610 so where the source term is e to the i omega t. 284 00:19:41,610 --> 00:19:45,960 And alternating, so this is electrical engineers 285 00:19:45,960 --> 00:19:50,950 would meet it constantly from alternating voltage source, 286 00:19:50,950 --> 00:19:56,610 alternating current source, AC, with frequency omega, 287 00:19:56,610 --> 00:19:58,590 60 cycles per second, for example. 288 00:20:02,400 --> 00:20:04,285 Why don't I just deal with this now? 289 00:20:06,900 --> 00:20:11,440 Because it involves complex numbers. 290 00:20:11,440 --> 00:20:17,810 And we've got to take a little step back and prepare for that. 291 00:20:17,810 --> 00:20:22,180 But when we do it, we'll get not only e 292 00:20:22,180 --> 00:20:25,910 to the i omega t, which I brought out, 293 00:20:25,910 --> 00:20:28,650 but also, it's real part. 294 00:20:28,650 --> 00:20:31,750 You remember the great formula with complex numbers, 295 00:20:31,750 --> 00:20:34,340 Euler's formula, that e to the i omega 296 00:20:34,340 --> 00:20:39,340 t is a combination of cosine omega t, the real part, 297 00:20:39,340 --> 00:20:44,150 and then the imaginary part is sine omega t. 298 00:20:44,150 --> 00:20:52,510 So this is looking like a complex problem. 299 00:20:52,510 --> 00:20:58,700 But it actually solves two real problems, cosine and sine. 300 00:20:58,700 --> 00:21:02,870 Cosine and sine will be on our short list of great functions 301 00:21:02,870 --> 00:21:04,480 that we can deal with. 302 00:21:04,480 --> 00:21:10,430 But to deal with them neatly, we need a little thought 303 00:21:10,430 --> 00:21:11,790 about complex numbers. 304 00:21:11,790 --> 00:21:15,330 So OK if I leave e to the i omega 305 00:21:15,330 --> 00:21:18,490 t for the end of the list, here? 306 00:21:21,440 --> 00:21:24,600 So I'm ready for another one, another source term. 307 00:21:24,600 --> 00:21:29,830 And I'm going to pick the step function. 308 00:21:29,830 --> 00:21:41,190 So the next example is going to be dy dt equals ay plus a step. 309 00:21:41,190 --> 00:21:48,680 Well, suppose I put H of t there. 310 00:21:48,680 --> 00:21:50,205 Suppose I put H of t. 311 00:21:52,820 --> 00:21:55,240 And I ask you for the solution to that guy. 312 00:21:57,820 --> 00:22:02,120 So that step function, its graph is here. 313 00:22:02,120 --> 00:22:06,450 It's 0 for negative time, and it's 1 for positive time. 314 00:22:09,520 --> 00:22:12,330 So we've already solved that problem, right? 315 00:22:12,330 --> 00:22:16,130 Where did I solve this equation? 316 00:22:16,130 --> 00:22:18,740 This equation is already on that board. 317 00:22:21,410 --> 00:22:22,190 Because why? 318 00:22:26,120 --> 00:22:30,470 Because H of t is for t positive. 319 00:22:30,470 --> 00:22:32,630 That's the only place we're looking. 320 00:22:32,630 --> 00:22:36,380 This whole problem, we're not looking at negative t. 321 00:22:36,380 --> 00:22:39,980 We're only looking at t from 0 forward. 322 00:22:39,980 --> 00:22:46,070 And what is H of t from 0 forward? 323 00:22:46,070 --> 00:22:46,600 It's 1. 324 00:22:46,600 --> 00:22:48,120 It's a constant. 325 00:22:48,120 --> 00:22:55,130 So that problem, as it stands, is identical to that problem. 326 00:22:55,130 --> 00:22:58,910 Same thing, we have a 1. 327 00:22:58,910 --> 00:23:01,020 No need to solve that again. 328 00:23:01,020 --> 00:23:09,970 The real example is when this function jumps up 329 00:23:09,970 --> 00:23:20,422 at some later time T. Now I have the function is H of t minus T. 330 00:23:20,422 --> 00:23:27,980 Do you see that, why the step function that jumps at time T 331 00:23:27,980 --> 00:23:30,340 has that formula? 332 00:23:30,340 --> 00:23:34,140 Because for little t before that time, 333 00:23:34,140 --> 00:23:39,120 in here, this is-- what's the deal? 334 00:23:39,120 --> 00:23:42,720 If little t is smaller than big T, 335 00:23:42,720 --> 00:23:50,880 then t minus T is negative, right? 336 00:23:50,880 --> 00:23:55,440 If t is in here, then t minus capital T 337 00:23:55,440 --> 00:23:56,950 is going to be a negative number. 338 00:23:56,950 --> 00:24:00,680 And H of a negative number is 0. 339 00:24:00,680 --> 00:24:04,980 But for t greater than capital T, this is a positive number. 340 00:24:04,980 --> 00:24:07,460 And H of a positive number is 1. 341 00:24:07,460 --> 00:24:11,340 Do you see how if you want to shift a graph, 342 00:24:11,340 --> 00:24:13,480 if you want the graph to shift, if you want 343 00:24:13,480 --> 00:24:19,570 to move the starting time, then algebraically, the way 344 00:24:19,570 --> 00:24:24,118 you do it is to change t to t minus the starting time. 345 00:24:26,522 --> 00:24:27,730 And that's what I want to do. 346 00:24:33,420 --> 00:24:40,010 So physically, what's happening with this equation? 347 00:24:40,010 --> 00:24:42,780 So it starts with y of 0 as before. 348 00:24:42,780 --> 00:24:46,910 Let's think of a bank balance and then other things, too. 349 00:24:46,910 --> 00:24:53,870 If it's a bank balance, we put in a certain amount, y of 0. 350 00:24:53,870 --> 00:24:54,860 We hope. 351 00:24:54,860 --> 00:24:57,710 And that grew. 352 00:24:57,710 --> 00:25:00,900 And then starting at time, capital T, 353 00:25:00,900 --> 00:25:04,850 this switch turns on. 354 00:25:04,850 --> 00:25:09,570 Actually, physically, step function 355 00:25:09,570 --> 00:25:15,300 is really often describing a switch that's turned on, now. 356 00:25:15,300 --> 00:25:20,190 This source term act begins to act at that time. 357 00:25:20,190 --> 00:25:21,590 And it acts at 1. 358 00:25:24,240 --> 00:25:29,320 So at time capital T we start putting money into our account. 359 00:25:29,320 --> 00:25:30,920 Or taking it out, of course. 360 00:25:30,920 --> 00:25:37,740 If this with a minus sign, I'd be putting money in. 361 00:25:37,740 --> 00:25:40,650 Sorry, I would start with some money in, y of 0. 362 00:25:44,290 --> 00:25:46,120 I would start with money in. 363 00:25:46,120 --> 00:25:49,080 Yeah, actually, tell me what's the solution 364 00:25:49,080 --> 00:25:55,560 to this equation that starts from y of 0? 365 00:25:55,560 --> 00:26:00,750 What's the solution up until the switch is turned on? 366 00:26:00,750 --> 00:26:05,300 What's the solution before this switch happens, 367 00:26:05,300 --> 00:26:08,640 this solution while this is still 0? 368 00:26:08,640 --> 00:26:11,870 So let's put that part of the answer down. 369 00:26:11,870 --> 00:26:17,275 This is for t smaller than T. What's the answer? 370 00:26:22,410 --> 00:26:23,620 This is all common sense. 371 00:26:23,620 --> 00:26:28,010 It's coming fast, so I'm asking these questions. 372 00:26:28,010 --> 00:26:32,730 And when I asked that question, it's a sort of indication 373 00:26:32,730 --> 00:26:36,180 that you can really see the answer. 374 00:26:36,180 --> 00:26:38,660 You don't need to go back to the textbook for that. 375 00:26:38,660 --> 00:26:40,070 What have we got here? 376 00:26:40,070 --> 00:26:40,570 Yeah? 377 00:26:40,570 --> 00:26:42,486 AUDIENCE: Is it the null solution [INAUDIBLE]? 378 00:26:42,486 --> 00:26:45,010 PROFESSOR: It'll be this guy. 379 00:26:45,010 --> 00:26:47,450 Yeah, the particular solution will be 0. 380 00:26:47,450 --> 00:26:52,850 Right, the particular solution is 0 before this is on. 381 00:26:52,850 --> 00:26:56,020 I'm sorry, the null solution is 0, 382 00:26:56,020 --> 00:26:58,980 and the particular solution, well, the particular solution 383 00:26:58,980 --> 00:27:01,640 is a guy that starts right. 384 00:27:01,640 --> 00:27:03,230 I don't know. 385 00:27:03,230 --> 00:27:07,370 Those names were not important. 386 00:27:07,370 --> 00:27:09,960 And then the question is-- so it's just 387 00:27:09,960 --> 00:27:11,790 our initial deposit growing. 388 00:27:15,920 --> 00:27:20,350 Now, all I ask, what about after time T? 389 00:27:20,350 --> 00:27:22,151 What about after time T? 390 00:27:25,540 --> 00:27:31,450 For t after time T, and hopefully, equal time T, 391 00:27:31,450 --> 00:27:33,470 what do you think y of t will be? 392 00:27:39,230 --> 00:27:43,120 Again, we want to separate in our minds 393 00:27:43,120 --> 00:27:48,810 the stuff that's starting from the initial condition 394 00:27:48,810 --> 00:27:54,410 from the stuff that's piling up because of the source. 395 00:27:54,410 --> 00:27:59,500 So one part will be that guy. 396 00:28:02,730 --> 00:28:04,730 I haven't given the complete answer. 397 00:28:04,730 --> 00:28:09,380 But this is continuing to grow. 398 00:28:09,380 --> 00:28:15,560 And because it's linear, we're always using this neat fact 399 00:28:15,560 --> 00:28:16,920 that our equation is linear. 400 00:28:16,920 --> 00:28:19,530 We can watch things separately, and then 401 00:28:19,530 --> 00:28:21,140 just add them together. 402 00:28:21,140 --> 00:28:23,770 So I plan to add this part, which 403 00:28:23,770 --> 00:28:29,520 comes from initial condition to a part 404 00:28:29,520 --> 00:28:35,880 that-- maybe we can guess it-- that's coming from the source. 405 00:28:35,880 --> 00:28:40,100 And how do we have any chance to guess it? 406 00:28:40,100 --> 00:28:45,770 Only because that particular source, once it's turned on, 407 00:28:45,770 --> 00:28:48,800 jumps to a constant 1, and we've solved 408 00:28:48,800 --> 00:28:51,710 the equation for a constant 1. 409 00:28:51,710 --> 00:28:52,910 Let me go back here. 410 00:28:56,030 --> 00:29:04,208 I think our answer to this question-- 411 00:29:04,208 --> 00:29:09,090 so this is like just first practice with a step function, 412 00:29:09,090 --> 00:29:12,750 to get the hang of a step function. 413 00:29:12,750 --> 00:29:17,100 So I'm seeing this same y of 0 e to the at in every case, 414 00:29:17,100 --> 00:29:20,970 because that's what happens to the initial deposit. 415 00:29:20,970 --> 00:29:23,720 I'll say grow, assuming the bank's paying 416 00:29:23,720 --> 00:29:25,900 a positive interest rate. 417 00:29:25,900 --> 00:29:29,940 And now, where did this term comes from? 418 00:29:29,940 --> 00:29:31,395 What did that term represent? 419 00:29:33,459 --> 00:29:35,000 AUDIENCE: The money that [INAUDIBLE]. 420 00:29:35,000 --> 00:29:36,390 PROFESSOR: The money that, yeah? 421 00:29:36,390 --> 00:29:38,015 AUDIENCE: They had each of [INAUDIBLE]. 422 00:29:38,015 --> 00:29:40,770 PROFESSOR: The money that came in and grew. 423 00:29:40,770 --> 00:29:45,090 It came in, and then it grew by itself, 424 00:29:45,090 --> 00:29:47,150 grew separately from that these guys. 425 00:29:47,150 --> 00:29:50,100 So the initial condition is growing along. 426 00:29:50,100 --> 00:29:52,650 And the money we put in starts growing. 427 00:29:52,650 --> 00:29:54,810 Now, the point is what? 428 00:29:54,810 --> 00:30:00,776 That over here, it's going to look just like that. 429 00:30:00,776 --> 00:30:03,870 So I'm going to have a 1 over a. 430 00:30:03,870 --> 00:30:06,690 And I'm going to have something like that. 431 00:30:06,690 --> 00:30:11,520 But can you just guess what's going to go in there? 432 00:30:11,520 --> 00:30:14,920 When I write it down, it'll make sense. 433 00:30:14,920 --> 00:30:20,410 So this term is representing what we have at time little t, 434 00:30:20,410 --> 00:30:24,080 later on, from the deposits we made, 435 00:30:24,080 --> 00:30:28,130 not the initial one, but the source, the continuing 436 00:30:28,130 --> 00:30:29,020 deposits. 437 00:30:29,020 --> 00:30:30,560 And let me write it. 438 00:30:30,560 --> 00:30:39,250 It's going to be a 1 over a e to the a something minus 1. 439 00:30:39,250 --> 00:30:42,500 It's going to look just like that guy. 440 00:30:42,500 --> 00:30:45,910 When I say that guy, let me point to it again-- 441 00:30:45,910 --> 00:30:48,440 e to the at minus 1. 442 00:30:48,440 --> 00:30:51,810 But it's not quite e to the at minus 1. 443 00:30:51,810 --> 00:30:52,520 What is it? 444 00:30:52,520 --> 00:30:53,770 AUDIENCE: t minus [INAUDIBLE]. 445 00:30:53,770 --> 00:31:03,560 PROFESSOR: t minus capital T, because it 446 00:31:03,560 --> 00:31:05,000 didn't start until that time. 447 00:31:08,900 --> 00:31:17,540 So I'm going to leave that as, like, reasonable, sensible. 448 00:31:20,550 --> 00:31:24,600 Think about a step function that's turned on a capital time 449 00:31:24,600 --> 00:31:27,200 T. Then it grows from that time. 450 00:31:27,200 --> 00:31:29,860 Of course, mentally, I never write down a formula 451 00:31:29,860 --> 00:31:35,900 like that without checking at t equal to T, 452 00:31:35,900 --> 00:31:39,040 because that's the one important point, at t equal capital 453 00:31:39,040 --> 00:31:43,940 T. What is this at t equal capital T? 454 00:31:43,940 --> 00:31:45,350 It's 0. 455 00:31:45,350 --> 00:31:49,680 At t equal capital T, this is e to the 0, which is 1 minus 1 456 00:31:49,680 --> 00:31:51,520 altogether 0. 457 00:31:51,520 --> 00:31:54,000 And is that the right answer? 458 00:31:54,000 --> 00:31:58,020 At t equal capital T is 0, should I have nothing here? 459 00:32:01,020 --> 00:32:01,520 Yes? 460 00:32:01,520 --> 00:32:02,020 No? 461 00:32:02,020 --> 00:32:04,190 Give me a head shake. 462 00:32:04,190 --> 00:32:07,350 Should I have nothing at t equal capital T? 463 00:32:07,350 --> 00:32:08,410 I've got nothing. 464 00:32:08,410 --> 00:32:12,120 e to the 0 minus 1, that's nothing? 465 00:32:12,120 --> 00:32:14,420 Yes, yes that's the right thing. 466 00:32:14,420 --> 00:32:21,770 Because at capital T, the source has just turned on, 467 00:32:21,770 --> 00:32:25,170 hasn't had time to build up anything, 468 00:32:25,170 --> 00:32:27,080 just that was the instant it turned on. 469 00:32:30,430 --> 00:32:32,900 So that's a step function. 470 00:32:32,900 --> 00:32:35,890 A step function is a little bit of a stretch 471 00:32:35,890 --> 00:32:40,420 from an ordinary function, but not as much 472 00:32:40,420 --> 00:32:44,860 of a stretch as its derivative. 473 00:32:44,860 --> 00:32:50,190 In a way, this is like the highlight for today, coming up, 474 00:32:50,190 --> 00:32:55,300 to deal with not only a step function, but a delta function. 475 00:33:01,490 --> 00:33:03,710 I guess every author and every teacher 476 00:33:03,710 --> 00:33:07,840 has to think am I going to let this delta function 477 00:33:07,840 --> 00:33:11,930 into my course or into the book? 478 00:33:11,930 --> 00:33:16,750 And my answer is yes. 479 00:33:16,750 --> 00:33:17,650 You have to do it. 480 00:33:17,650 --> 00:33:19,050 You should do it. 481 00:33:19,050 --> 00:33:24,800 Delta functions are-- they're not true functions. 482 00:33:24,800 --> 00:33:27,480 As we'll see, no true function can 483 00:33:27,480 --> 00:33:29,600 do what a delta function does. 484 00:33:29,600 --> 00:33:35,170 But it's such an intuitive, fantastic model 485 00:33:35,170 --> 00:33:39,760 of things happening over a very, very short time. 486 00:33:39,760 --> 00:33:42,440 We just make that short time into 0. 487 00:33:42,440 --> 00:33:45,400 So we're saying with the delta function, 488 00:33:45,400 --> 00:33:53,110 we're going to say that something can happen in 0 time. 489 00:33:55,620 --> 00:33:58,020 Something can happen in 0 time. 490 00:33:58,020 --> 00:34:03,950 It's a model of, you know, when a bat hits a ball. 491 00:34:03,950 --> 00:34:06,520 There's a very short time. 492 00:34:06,520 --> 00:34:08,610 Or a golf club hits a golf ball. 493 00:34:08,610 --> 00:34:11,170 There's a very short time interval 494 00:34:11,170 --> 00:34:12,194 when they're in contact. 495 00:34:14,969 --> 00:34:20,170 We're modeling that by 0 time, but still, the ball 496 00:34:20,170 --> 00:34:23,050 gets an impulse. 497 00:34:23,050 --> 00:34:28,940 Normally, for 0 time, if you're doing things continuously, 498 00:34:28,940 --> 00:34:31,790 what you do over 0 time is no importance. 499 00:34:31,790 --> 00:34:34,389 But we're not doing things continuously, at all. 500 00:34:34,389 --> 00:34:36,960 So here we go. 501 00:34:36,960 --> 00:34:40,929 You've seen this guy, I think. 502 00:34:40,929 --> 00:34:44,449 But if you haven't, here's the time to see it. 503 00:34:44,449 --> 00:34:48,360 So the delta function is the derivative 504 00:34:48,360 --> 00:34:53,219 of-- so I've written three important functions up here. 505 00:34:53,219 --> 00:34:57,210 Let me start with a continuous one. 506 00:34:57,210 --> 00:35:05,090 That function, the ramp is 0, and then the ramp 507 00:35:05,090 --> 00:35:06,810 suddenly ramps up to t. 508 00:35:11,160 --> 00:35:13,070 Take its derivative. 509 00:35:13,070 --> 00:35:16,570 So the derivative, the slope of the ramp function 510 00:35:16,570 --> 00:35:18,340 is certainly 0 there. 511 00:35:18,340 --> 00:35:20,220 And here, the slope is 1. 512 00:35:20,220 --> 00:35:23,760 So the slope jumped from 0 to 1. 513 00:35:23,760 --> 00:35:31,190 The slope of the ramp function is the step function. 514 00:35:31,190 --> 00:35:34,640 Derivative of ramp equals step. 515 00:35:34,640 --> 00:35:36,340 Why don't I write those words down? 516 00:35:40,350 --> 00:35:45,790 Derivative of ramp equals step. 517 00:35:50,040 --> 00:35:53,345 So there is already the step function. 518 00:35:56,332 --> 00:36:00,240 In pure calculus, the step function 519 00:36:00,240 --> 00:36:04,190 has already got a little question mark. 520 00:36:04,190 --> 00:36:10,410 Because at that point, the derivative in a calculus course 521 00:36:10,410 --> 00:36:13,260 doesn't exist, strictly doesn't exist, 522 00:36:13,260 --> 00:36:15,500 because we get a different answer 523 00:36:15,500 --> 00:36:21,540 0 on the left side from the answer, 1 on the right side. 524 00:36:21,540 --> 00:36:23,670 We just go with that. 525 00:36:23,670 --> 00:36:25,600 I'm not going to worry about what 526 00:36:25,600 --> 00:36:28,780 is its value at that point. 527 00:36:28,780 --> 00:36:33,570 It's 0 up for t negative, and it's 1 for t positive. 528 00:36:33,570 --> 00:36:36,720 And often, I'll take it 1 for t equals 0, also. 529 00:36:36,720 --> 00:36:39,590 Usually, I will. 530 00:36:39,590 --> 00:36:43,460 That's the small problem. 531 00:36:43,460 --> 00:36:48,020 Now, the bigger problem is the derivative of the-- so this 532 00:36:48,020 --> 00:36:52,910 is now the derivative of the step function. 533 00:36:52,910 --> 00:36:55,300 So what's the derivative of this step function? 534 00:36:55,300 --> 00:36:58,630 Well, the derivative along there is certainly 0. 535 00:36:58,630 --> 00:37:02,050 The derivative along here is certainly 0. 536 00:37:02,050 --> 00:37:07,250 But the derivative, when that jumped, 537 00:37:07,250 --> 00:37:10,930 the derivative, the slope was infinite. 538 00:37:10,930 --> 00:37:12,500 That line is vertical. 539 00:37:12,500 --> 00:37:14,090 Its slope is infinite. 540 00:37:14,090 --> 00:37:20,520 So at that one point, you have an affinity, here, delta of 0. 541 00:37:20,520 --> 00:37:22,510 You could say delta of 0 is infinite. 542 00:37:26,390 --> 00:37:29,640 But you haven't said much, there. 543 00:37:29,640 --> 00:37:34,130 Infinite is too vague. 544 00:37:34,130 --> 00:37:37,280 Actually, I wouldn't know if you gave 545 00:37:37,280 --> 00:37:39,410 me infinite or 2 times infinite. 546 00:37:39,410 --> 00:37:41,420 I couldn't tell the difference. 547 00:37:41,420 --> 00:37:48,240 So I'll put it in quotes, because it sort of gives us 548 00:37:48,240 --> 00:37:48,850 comfort. 549 00:37:48,850 --> 00:37:51,390 But it doesn't mean much. 550 00:37:51,390 --> 00:37:53,060 What does mean much? 551 00:37:53,060 --> 00:37:54,470 Somehow that's important. 552 00:37:58,530 --> 00:38:00,790 Can I tell you how to work with delta functions, 553 00:38:00,790 --> 00:38:02,340 how to think about delta functions? 554 00:38:05,660 --> 00:38:08,590 It's the right way to think about delta function. 555 00:38:08,590 --> 00:38:11,990 So here's some comment on delta function. 556 00:38:18,990 --> 00:38:22,370 Giving the values of the function, 0, and infinity, 557 00:38:22,370 --> 00:38:26,690 and 0, is not the best. 558 00:38:26,690 --> 00:38:30,050 What you can do with a delta function 559 00:38:30,050 --> 00:38:33,420 is you can integrate it. 560 00:38:33,420 --> 00:38:38,380 You can define the function by integrals. 561 00:38:38,380 --> 00:38:40,340 Integrals of things are nice. 562 00:38:40,340 --> 00:38:44,200 Do you think in your mind when you take derivatives, 563 00:38:44,200 --> 00:38:48,450 as we did going left to right, we were taking derivatives. 564 00:38:48,450 --> 00:38:51,650 The function was getting crazy. 565 00:38:51,650 --> 00:38:57,570 When we go right to left, take integrals, those are smoothing. 566 00:38:57,570 --> 00:39:00,160 Integrals make functions smoother. 567 00:39:00,160 --> 00:39:03,090 They cancel noise. 568 00:39:03,090 --> 00:39:05,210 They smooth the function out. 569 00:39:05,210 --> 00:39:09,320 So what we can do is to take the integral of the delta function. 570 00:39:11,920 --> 00:39:16,900 We could take it from any negative number 571 00:39:16,900 --> 00:39:19,890 to any positive number. 572 00:39:19,890 --> 00:39:23,390 And what answer would we get? 573 00:39:23,390 --> 00:39:27,800 What would be the right, well, the one thing people 574 00:39:27,800 --> 00:39:29,990 know about the delta function is-- 575 00:39:29,990 --> 00:39:33,750 and actually, it's the key thing-- the integral 576 00:39:33,750 --> 00:39:36,900 of the delta function. 577 00:39:36,900 --> 00:39:40,910 Again, I'm integrating the delta function 578 00:39:40,910 --> 00:39:45,150 from some negative number up to some positive number. 579 00:39:45,150 --> 00:39:48,930 And it doesn't matter where n is, because the function is 0 580 00:39:48,930 --> 00:39:49,780 there and there. 581 00:39:49,780 --> 00:39:53,240 But what's the answer here? 582 00:39:53,240 --> 00:39:54,430 Put me out of my misery. 583 00:39:54,430 --> 00:39:57,440 Just tell me the number I'm looking for, here, 584 00:39:57,440 --> 00:39:58,980 the integral of the delta function. 585 00:39:58,980 --> 00:40:00,504 Or maybe you haven't met it. 586 00:40:00,504 --> 00:40:01,420 AUDIENCE: [INAUDIBLE]. 587 00:40:01,420 --> 00:40:03,640 PROFESSOR: It's? 588 00:40:03,640 --> 00:40:07,510 It's the one good number you could guess. 589 00:40:07,510 --> 00:40:09,160 It's 1. 590 00:40:09,160 --> 00:40:12,280 Now, why is it 1? 591 00:40:12,280 --> 00:40:16,380 Because if the delta function is the derivative of the step 592 00:40:16,380 --> 00:40:23,470 function, this should be the step function evaluated between 593 00:40:23,470 --> 00:40:29,000 N and P. This should be the step function, , here, 594 00:40:29,000 --> 00:40:30,960 minus the step function, there 595 00:40:30,960 --> 00:40:34,472 And what is the step function? 596 00:40:34,472 --> 00:40:35,680 You have to keep it straight. 597 00:40:35,680 --> 00:40:37,420 Am I talking about the delta function? 598 00:40:37,420 --> 00:40:42,700 No, right now, I've integrated it to get H of t. 599 00:40:42,700 --> 00:40:46,410 So this is H of P at the positive side, 600 00:40:46,410 --> 00:40:52,640 minus H of N. That's what integration's about. 601 00:40:52,640 --> 00:40:54,960 And what do I get? 602 00:40:54,960 --> 00:41:00,440 1, because H of P, the step function here, H is 1. 603 00:41:00,440 --> 00:41:02,670 And here, it's 0, so I get 1. 604 00:41:08,310 --> 00:41:11,220 Good, that's the thing that everybody 605 00:41:11,220 --> 00:41:13,690 remembers about the delta function. 606 00:41:13,690 --> 00:41:18,860 And now I can make sense out of 2 delta function, 2 delta of t. 607 00:41:18,860 --> 00:41:19,900 That could be my source. 608 00:41:22,580 --> 00:41:26,930 So if 2 delta of t was my source, 609 00:41:26,930 --> 00:41:29,240 what's the graph of 2 delta of t? 610 00:41:29,240 --> 00:41:33,820 Again, it's 0 infinite 0. 611 00:41:33,820 --> 00:41:38,470 You really can't tell from the infinity what's up, 612 00:41:38,470 --> 00:41:43,820 but what would be the integral of 2 delta of t, 613 00:41:43,820 --> 00:41:47,230 the integral of 2 delta of t or some other? 614 00:41:47,230 --> 00:41:49,340 Well, let me put in the 2, here? 615 00:41:49,340 --> 00:41:53,290 What's the integral of 2 delta of t, would be 2H of t. 616 00:41:53,290 --> 00:41:53,840 Keep going. 617 00:41:53,840 --> 00:41:54,870 What do I get here? 618 00:41:54,870 --> 00:41:55,400 AUDIENCE: 2. 619 00:41:55,400 --> 00:41:57,880 PROFESSOR: It would be 2 of these guys, 2 620 00:41:57,880 --> 00:41:59,851 of these, 2 of these, 2. 621 00:41:59,851 --> 00:42:00,350 All right? 622 00:42:03,990 --> 00:42:09,120 So we made sense out of the strength of the impulse, 623 00:42:09,120 --> 00:42:13,260 how hard the bat hit the ball. 624 00:42:13,260 --> 00:42:16,320 But of course, we need units in there. 625 00:42:16,320 --> 00:42:18,700 We have to have units. 626 00:42:18,700 --> 00:42:25,650 And here, the value for that unit was 2. 627 00:42:25,650 --> 00:42:29,780 Now, I'm going to-- because this is really worth 628 00:42:29,780 --> 00:42:32,990 doing with delta functions. 629 00:42:32,990 --> 00:42:37,290 I didn't ask at the start have you seen them before. 630 00:42:37,290 --> 00:42:40,720 But they are worth seeing. 631 00:42:40,720 --> 00:42:42,470 And they just take a little practice. 632 00:42:42,470 --> 00:42:45,500 But then in the end, delta functions 633 00:42:45,500 --> 00:42:49,950 are way easier to work with than some complicated function that 634 00:42:49,950 --> 00:42:54,880 attempts to model this. 635 00:42:54,880 --> 00:43:01,320 We could model that by some Gaussian curve or something. 636 00:43:01,320 --> 00:43:05,330 All the integrations would become impossible right away. 637 00:43:05,330 --> 00:43:12,040 We could model it by a step function up 638 00:43:12,040 --> 00:43:13,890 and a step function down. 639 00:43:13,890 --> 00:43:17,920 Then the integrations would be possible. 640 00:43:17,920 --> 00:43:22,360 But still, we have this finite width. 641 00:43:22,360 --> 00:43:25,230 I could let that width go to 0 and let 642 00:43:25,230 --> 00:43:27,444 the height go to infinity. 643 00:43:27,444 --> 00:43:28,360 And what would happen? 644 00:43:28,360 --> 00:43:30,590 I'd get the delta function. 645 00:43:30,590 --> 00:43:33,930 So that's one way to create a delta function, if you like. 646 00:43:33,930 --> 00:43:36,570 If you're OK with step functions, 647 00:43:36,570 --> 00:43:41,000 then one way to create delta is to take a big step up, step 648 00:43:41,000 --> 00:43:45,320 down, and then let the size of the step grow 649 00:43:45,320 --> 00:43:47,960 and the width of the steps shrink. 650 00:43:47,960 --> 00:43:52,990 Keep the area 1, because area is integral. 651 00:43:52,990 --> 00:43:56,360 So I keep this, that little width, 652 00:43:56,360 --> 00:43:58,580 times this big height equal to 1. 653 00:43:58,580 --> 00:44:02,110 And in the end, I get delta. 654 00:44:02,110 --> 00:44:08,610 Now again, my point is that delta functions, 655 00:44:08,610 --> 00:44:10,190 that you really understand them. 656 00:44:10,190 --> 00:44:13,320 What you can legitimately do with them is integrate them. 657 00:44:13,320 --> 00:44:19,090 But now in later problems, we might have not a 1 or a 2, 658 00:44:19,090 --> 00:44:27,570 but a function in here, like cosine t, or e to the t, 659 00:44:27,570 --> 00:44:30,380 or q of t. 660 00:44:30,380 --> 00:44:32,560 Can I practice with those? 661 00:44:32,560 --> 00:44:35,100 Can I put in a function f of t? 662 00:44:35,100 --> 00:44:37,640 I didn't leave enough space to write f of t, 663 00:44:37,640 --> 00:44:47,090 so I'm going to put it in here. f of t delta of t dt. 664 00:44:47,090 --> 00:44:50,873 And I'm going to go for the answer, there. 665 00:44:57,780 --> 00:45:01,390 My question is what does that equal? 666 00:45:01,390 --> 00:45:03,900 You see what the question is? 667 00:45:03,900 --> 00:45:08,100 I got my delta function, which I only just met. 668 00:45:08,100 --> 00:45:11,970 And I'm multiplying it by some ordinary function. 669 00:45:11,970 --> 00:45:15,040 f of t gives us no problems. 670 00:45:15,040 --> 00:45:16,250 Think of cosine t. 671 00:45:16,250 --> 00:45:19,140 Think of e to the t. 672 00:45:19,140 --> 00:45:23,697 What do you think is the right answer for that? 673 00:45:23,697 --> 00:45:25,280 What do you think is the right answer? 674 00:45:25,280 --> 00:45:28,350 And this tells you what the delta function 675 00:45:28,350 --> 00:45:29,960 is when you see this. 676 00:45:35,450 --> 00:45:39,083 What do I need to know about f of t to get an answer, here? 677 00:45:42,540 --> 00:45:48,030 Do I need to know what f is at t equals minus 1? 678 00:45:48,030 --> 00:45:51,100 You could see from the way my voice asked 679 00:45:51,100 --> 00:45:55,220 that question that the answer is no. 680 00:45:55,220 --> 00:45:59,830 Why do I not care what f is at minus 1? 681 00:45:59,830 --> 00:46:00,330 Yeah? 682 00:46:00,330 --> 00:46:01,310 AUDIENCE: Because you're multiplying by [INAUDIBLE]. 683 00:46:01,310 --> 00:46:02,726 PROFESSOR: Because I'm multiplying 684 00:46:02,726 --> 00:46:04,940 by somebody that's 0. 685 00:46:04,940 --> 00:46:09,890 And similarly, at f equal minus 1/2, or at f equal plus 1/3, 686 00:46:09,890 --> 00:46:11,730 all those f's make no difference, 687 00:46:11,730 --> 00:46:17,250 because they're all multiplying 0. 688 00:46:17,250 --> 00:46:18,720 What does make a difference? 689 00:46:18,720 --> 00:46:24,680 What's the key information about f that does 690 00:46:24,680 --> 00:46:27,220 come into the answer? 691 00:46:27,220 --> 00:46:28,910 f at? 692 00:46:28,910 --> 00:46:32,154 At just at that one point, f at? 693 00:46:32,154 --> 00:46:33,420 AUDIENCE: [INAUDIBLE] 694 00:46:33,420 --> 00:46:37,350 PROFESSOR: 0, f at 0 is the action. 695 00:46:37,350 --> 00:46:38,996 The impulse is happening. 696 00:46:38,996 --> 00:46:40,120 The bat's hitting the ball. 697 00:46:43,980 --> 00:46:46,700 So we're modeling rocket launching, here. 698 00:46:46,700 --> 00:46:52,640 We're launching in 0 seconds instead of a finite time. 699 00:46:52,640 --> 00:46:57,020 So in other words, well, I don't know 700 00:46:57,020 --> 00:47:02,130 how to put this answer down other than just to write it. 701 00:47:02,130 --> 00:47:04,910 I guess I'm hoping you're with me 702 00:47:04,910 --> 00:47:06,930 in seeing that what it should be. 703 00:47:10,760 --> 00:47:12,440 Can I just write it? 704 00:47:12,440 --> 00:47:15,690 All that matters is what f is at t 705 00:47:15,690 --> 00:47:19,670 equals 0, because that's where all the action is. 706 00:47:19,670 --> 00:47:24,170 And that f of 0, if f of 0 was the 2 707 00:47:24,170 --> 00:47:28,280 that I had there a little while ago, then the answer will be 2. 708 00:47:28,280 --> 00:47:34,880 If f of 0 is a 1, if the answer is f of 0 times 709 00:47:34,880 --> 00:47:38,430 1-- and I won't write times 1. 710 00:47:38,430 --> 00:47:39,180 That's ridiculous. 711 00:47:43,880 --> 00:47:46,520 Now we can integrate delta functions, not just 712 00:47:46,520 --> 00:47:50,390 a single integral of delta, but integral 713 00:47:50,390 --> 00:47:53,800 of a function, a nice function times delta. 714 00:47:53,800 --> 00:47:55,330 And we get f of 0. 715 00:47:55,330 --> 00:48:00,110 So can I just, while we're on the subject of delta functions, 716 00:48:00,110 --> 00:48:02,020 ask you a few examples? 717 00:48:02,020 --> 00:48:11,635 What is the integral of e to the t delta of t dt? 718 00:48:11,635 --> 00:48:12,431 AUDIENCE: It's 1. 719 00:48:12,431 --> 00:48:13,680 PROFESSOR: Yeah, say it again? 720 00:48:13,680 --> 00:48:14,430 AUDIENCE: It's 1. 721 00:48:14,430 --> 00:48:15,440 PROFESSOR: It's 1. 722 00:48:15,440 --> 00:48:16,920 It's 1, right. 723 00:48:16,920 --> 00:48:21,340 Because e to the t, at the only point we care about, t equal 0 724 00:48:21,340 --> 00:48:23,740 is 1. 725 00:48:23,740 --> 00:48:26,310 And what if I change that to sine t? 726 00:48:30,480 --> 00:48:34,630 Suppose I integrate sine t times delta of t? 727 00:48:34,630 --> 00:48:37,580 What do I get now? 728 00:48:37,580 --> 00:48:39,030 I get? 729 00:48:39,030 --> 00:48:39,530 AUDIENCE: 0. 730 00:48:39,530 --> 00:48:40,550 PROFESSOR: 0, right. 731 00:48:40,550 --> 00:48:43,080 And actually, that's totally reasonable. 732 00:48:43,080 --> 00:48:47,600 This is a function, which is yeah, it's an odd function. 733 00:48:51,190 --> 00:48:57,010 Anyway, sine, if I switch t to negative t, it goes negative. 734 00:48:57,010 --> 00:48:59,310 0 is the right answer. 735 00:48:59,310 --> 00:49:00,670 Let me ask you this one. 736 00:49:00,670 --> 00:49:02,730 What about delta of t squared? 737 00:49:07,190 --> 00:49:11,456 Because if we're up for a delta function, we might square it. 738 00:49:13,980 --> 00:49:16,320 Now we've got a high-powered function, 739 00:49:16,320 --> 00:49:20,480 because squaring this crazy function delta of t 740 00:49:20,480 --> 00:49:23,170 gives us something truly crazy. 741 00:49:23,170 --> 00:49:27,604 And what answer would you expect for that? 742 00:49:27,604 --> 00:49:28,463 AUDIENCE: 1. 743 00:49:28,463 --> 00:49:29,713 PROFESSOR: Would you expect 1? 744 00:49:33,790 --> 00:49:35,990 So this is like? 745 00:49:35,990 --> 00:49:38,350 I'm just getting intuition working, here, 746 00:49:38,350 --> 00:49:40,801 for delta functions. 747 00:49:40,801 --> 00:49:41,550 What do you think? 748 00:49:41,550 --> 00:49:44,505 I'm looking at the energy when I square something. 749 00:49:49,140 --> 00:49:50,747 OK, so we had a guess of 1. 750 00:49:50,747 --> 00:49:51,705 Is there another guess? 751 00:49:54,530 --> 00:49:55,030 Yeah? 752 00:49:55,030 --> 00:49:55,780 AUDIENCE: A third? 753 00:49:55,780 --> 00:49:56,562 PROFESSOR: Sorry? 754 00:49:56,562 --> 00:49:57,460 AUDIENCE: 1/3. 755 00:49:57,460 --> 00:50:01,130 PROFESSOR: 1/3, that's our second guess. 756 00:50:01,130 --> 00:50:10,090 I'm open for other guesses before I-- OK, we 757 00:50:10,090 --> 00:50:14,770 have a rule here for f of t. 758 00:50:14,770 --> 00:50:20,120 And now what is the f of t that I'm asking about in this case? 759 00:50:20,120 --> 00:50:22,150 It's delta of t, right? 760 00:50:22,150 --> 00:50:27,510 If f of t is delta of t, then that would match this. 761 00:50:27,510 --> 00:50:30,886 And therefore, the answer should match. 762 00:50:30,886 --> 00:50:32,510 Do you see what I'm shooting for, yeah? 763 00:50:32,510 --> 00:50:33,140 AUDIENCE: It'd be infinity? 764 00:50:33,140 --> 00:50:34,405 PROFESSOR: It'd be infinity. 765 00:50:34,405 --> 00:50:35,970 It would be infinity. 766 00:50:35,970 --> 00:50:44,650 That's delta of t squared is that's an infinite energy 767 00:50:44,650 --> 00:50:45,150 function. 768 00:50:45,150 --> 00:50:47,980 You never meet it, actually. 769 00:50:47,980 --> 00:50:50,570 I apologize, so so write it down there. 770 00:50:50,570 --> 00:50:53,600 I could erase it right away because you basically 771 00:50:53,600 --> 00:50:55,000 never see it. 772 00:50:55,000 --> 00:50:57,260 It's infinite energy. 773 00:50:57,260 --> 00:50:59,220 Well, I think you'd see it. 774 00:50:59,220 --> 00:51:03,060 I mean, we're really going back to the days of Norbert Wiener. 775 00:51:03,060 --> 00:51:05,060 When I came to the math department, 776 00:51:05,060 --> 00:51:09,210 Norbert Wiener was still here, still alive, still 777 00:51:09,210 --> 00:51:17,100 walking the hallway by touching the wall and counting offices. 778 00:51:17,100 --> 00:51:23,310 And hard to talk to, because he always had a lot to say. 779 00:51:23,310 --> 00:51:26,930 And you got kind of allowed to listen. 780 00:51:26,930 --> 00:51:32,250 So anyway, Wiener was among the first 781 00:51:32,250 --> 00:51:37,450 to really use delta functions, successfully 782 00:51:37,450 --> 00:51:38,380 use delta functions. 783 00:51:38,380 --> 00:51:42,320 Anyway, this is the big one. 784 00:51:42,320 --> 00:51:43,178 This is the big one. 785 00:51:47,600 --> 00:51:49,820 Now, so what's all that about? 786 00:51:49,820 --> 00:51:55,820 I guess I was trying to prepare by talking 787 00:51:55,820 --> 00:52:01,730 about this function prepare for the equation 788 00:52:01,730 --> 00:52:04,390 when that's the source. 789 00:52:04,390 --> 00:52:09,532 So dy equal ay plus a delta function. 790 00:52:13,900 --> 00:52:19,840 Let me bring that delta function in at time T. 791 00:52:19,840 --> 00:52:22,460 So how do you interpret that equation? 792 00:52:22,460 --> 00:52:24,440 So like part of this morning's lecture 793 00:52:24,440 --> 00:52:30,450 is to get a first handle on an impulse. 794 00:52:30,450 --> 00:52:33,430 So let me write that word impulse, here. 795 00:52:36,940 --> 00:52:39,110 Where am I going to write it? 796 00:52:39,110 --> 00:52:45,490 So delta is an impulse. 797 00:52:45,490 --> 00:52:48,200 That's our ordinary English word for something 798 00:52:48,200 --> 00:52:49,920 that happens fast. 799 00:52:49,920 --> 00:52:53,660 And y of t is the impulse response. 800 00:53:05,260 --> 00:53:09,130 And this is the most important. 801 00:53:13,110 --> 00:53:16,020 Well, I said e to the st was the most important. 802 00:53:16,020 --> 00:53:18,830 How can I have two most important examples? 803 00:53:18,830 --> 00:53:21,670 Well, they're a tie, let's say. 804 00:53:21,670 --> 00:53:26,260 e to the st is the most important ordinary function. 805 00:53:26,260 --> 00:53:30,610 It's the key to the whole course. 806 00:53:30,610 --> 00:53:36,450 Delta of t, the impulse, is the important one 807 00:53:36,450 --> 00:53:40,030 because if I can solve it for a delta function, 808 00:53:40,030 --> 00:53:42,690 I can solve it for anything. 809 00:53:42,690 --> 00:53:48,080 Let's see if we can solve it for a delta function, 810 00:53:48,080 --> 00:53:53,390 a delta function, an impulse that starts at time T. Again, 811 00:53:53,390 --> 00:53:57,620 I'm just going to start writing down the solution 812 00:53:57,620 --> 00:54:01,890 and ask for your help what to write next. 813 00:54:01,890 --> 00:54:06,550 So what do you expect as a first term in the solution? 814 00:54:06,550 --> 00:54:08,980 So I'm starting again from y of 0. 815 00:54:13,780 --> 00:54:18,070 Let's see if we can solve it by common sense. 816 00:54:22,530 --> 00:54:25,850 So how do I start the solution to this? 817 00:54:28,410 --> 00:54:31,070 Everybody sees what this equation is saying. 818 00:54:31,070 --> 00:54:36,260 I have an initial deposit of y of 0 that starts growing. 819 00:54:36,260 --> 00:54:39,910 And then at time capital T I make a deposit. 820 00:54:39,910 --> 00:54:45,980 At that moment, at that instant, I make a deposit of 1. 821 00:54:45,980 --> 00:54:49,640 That's an instant deposit of 1. 822 00:54:49,640 --> 00:54:51,530 Which is, of course, what I do in reality. 823 00:54:51,530 --> 00:54:53,250 I take $1 to the bank. 824 00:54:53,250 --> 00:54:54,420 They've got it now. 825 00:54:54,420 --> 00:54:59,290 At time T, I give them that $1, and it starts earning interest. 826 00:54:59,290 --> 00:55:01,820 So what about y of t? 827 00:55:01,820 --> 00:55:02,570 What do you think? 828 00:55:02,570 --> 00:55:06,355 What's the first term coming from y of 0? 829 00:55:09,750 --> 00:55:11,930 So the term coming from y of 0 will 830 00:55:11,930 --> 00:55:15,160 be y of 0 to start with, e to at. 831 00:55:20,060 --> 00:55:23,460 That takes care of the y of 0. 832 00:55:23,460 --> 00:55:24,580 Now, I need something. 833 00:55:27,640 --> 00:55:34,310 It's like this, plus I need something 834 00:55:34,310 --> 00:55:43,010 that accounts for what this deposit brings. 835 00:55:43,010 --> 00:55:47,940 So up until time T, what do I put? 836 00:55:47,940 --> 00:55:54,550 So this is for t smaller than T and t bigger than T. 837 00:55:54,550 --> 00:55:58,080 So what goes there? 838 00:55:58,080 --> 00:56:08,330 For t smaller than T, what's the benefit 839 00:56:08,330 --> 00:56:11,320 from the delta function? 840 00:56:11,320 --> 00:56:15,530 0, didn't happen yet. 841 00:56:15,530 --> 00:56:18,810 For t bigger than T, what's the benefit 842 00:56:18,810 --> 00:56:20,360 from the delta function? 843 00:56:20,360 --> 00:56:22,460 AUDIENCE: [INAUDIBLE]. 844 00:56:22,460 --> 00:56:26,470 PROFESSOR: For t bigger than T, well, that's right. 845 00:56:26,470 --> 00:56:28,170 OK, but now I've made that deposit 846 00:56:28,170 --> 00:56:34,570 at time capital T. Whatever's going 847 00:56:34,570 --> 00:56:40,800 there is whatever I'm getting from that deposit. 848 00:56:40,800 --> 00:56:45,850 At time capital T, I gave them $1, 849 00:56:45,850 --> 00:56:49,140 and they start paying interest on it. 850 00:56:49,140 --> 00:56:50,748 What's going to go there? 851 00:56:56,120 --> 00:57:01,050 So if I gave them $1 at that initial time, so that $1 852 00:57:01,050 --> 00:57:03,910 would have been part of y of 0. 853 00:57:03,910 --> 00:57:07,160 What did I get at a later time? 854 00:57:07,160 --> 00:57:09,390 e to the at. 855 00:57:09,390 --> 00:57:11,860 Now I'm waiting. 856 00:57:11,860 --> 00:57:15,740 I'm giving them the dollar at time capital T, 857 00:57:15,740 --> 00:57:18,220 and it starts growing. 858 00:57:18,220 --> 00:57:20,930 So what do I have at a later time, 859 00:57:20,930 --> 00:57:24,240 for t later than capital T? 860 00:57:24,240 --> 00:57:26,040 What has that $1 grown into? 861 00:57:28,690 --> 00:57:36,350 e to the a times the-- right, it's critical. 862 00:57:36,350 --> 00:57:37,760 It's the elapsed time. 863 00:57:37,760 --> 00:57:39,830 It's the time since the deposit. 864 00:57:39,830 --> 00:57:41,240 Is that right? 865 00:57:41,240 --> 00:57:42,414 So what do I put here? 866 00:57:42,414 --> 00:57:43,580 AUDIENCE: t minus capital T? 867 00:57:43,580 --> 00:57:45,570 PROFESSOR: t minus capital T, good. 868 00:57:48,690 --> 00:57:56,780 Apologies to bug you about this, but the only way 869 00:57:56,780 --> 00:58:03,310 to learn this stuff from a lecture is to be part of it. 870 00:58:03,310 --> 00:58:07,920 So I constantly ask you, instead of just writing down a formula. 871 00:58:07,920 --> 00:58:10,560 I think that looks good. 872 00:58:10,560 --> 00:58:18,630 So suddenly, what does this amount to at t equal capital T? 873 00:58:18,630 --> 00:58:21,490 Maybe I should allow t equal capital T. 874 00:58:21,490 --> 00:58:24,010 At t equal capital T, what do I have here? 875 00:58:24,010 --> 00:58:24,840 AUDIENCE: 1. 876 00:58:24,840 --> 00:58:25,830 PROFESSOR: 1. 877 00:58:25,830 --> 00:58:27,700 That's my $1. 878 00:58:27,700 --> 00:58:31,160 At t equal capital T, we've got $1. 879 00:58:31,160 --> 00:58:32,220 And later it's grown. 880 00:58:34,900 --> 00:58:37,210 So we have now solved. 881 00:58:37,210 --> 00:58:38,930 We have found the impulse response. 882 00:58:41,490 --> 00:58:43,290 We have found the impulse response 883 00:58:43,290 --> 00:58:52,820 when the impulse happened at capital T. That was good going. 884 00:58:52,820 --> 00:59:02,220 Now, I've given you my list of examples 885 00:59:02,220 --> 00:59:08,220 with the pause on the sine and cosine. 886 00:59:08,220 --> 00:59:15,040 I pause on the sine and cosine because one way 887 00:59:15,040 --> 00:59:18,690 to think about sine and cosine is to get into complex numbers. 888 00:59:18,690 --> 00:59:23,618 And that's really for next time. 889 00:59:27,610 --> 00:59:31,170 But apart from that, we've done all the examples, 890 00:59:31,170 --> 00:59:33,500 so are we ready? 891 00:59:33,500 --> 00:59:38,280 Oh yeah, I'm going to try for the big thing, the big formula. 892 00:59:38,280 --> 00:59:41,870 So this is the key result of section 1.4, 893 00:59:41,870 --> 00:59:44,470 the solution to this equation. 894 00:59:44,470 --> 00:59:46,555 So I'm going back to the original equation. 895 00:59:51,410 --> 00:59:57,020 And just see if we can write down a formula for the answer. 896 00:59:57,020 --> 00:59:59,460 So let me write the equation again. 897 00:59:59,460 --> 01:00:04,280 dy dt is ay plus some source. 898 01:00:08,280 --> 01:00:12,106 I think we can write down a formula that looks right. 899 01:00:16,210 --> 01:00:19,030 And we could then actually plug it in and see, yeah, 900 01:00:19,030 --> 01:00:21,010 it is right. 901 01:00:21,010 --> 01:00:23,710 So what's going to go into this formula? 902 01:00:23,710 --> 01:00:28,870 We got enough examples, so now let's go for the whole thing. 903 01:00:28,870 --> 01:00:38,710 So y of t, first of all, comes whatever 904 01:00:38,710 --> 01:00:40,562 depends on the initial condition. 905 01:00:44,210 --> 01:00:48,280 So how much do we have at a later time when 906 01:00:48,280 --> 01:00:52,480 our initial deposit was y of 0? 907 01:00:52,480 --> 01:00:56,430 So that's the one we've seen in every example. 908 01:00:56,430 --> 01:01:01,560 Every one of these things has this term growing out 909 01:01:01,560 --> 01:01:03,150 of y of 0. 910 01:01:03,150 --> 01:01:06,190 So let me put that in again. 911 01:01:06,190 --> 01:01:13,460 So the part that grows out of y of 0 is y of 0 e to the at. 912 01:01:13,460 --> 01:01:14,380 That's OK. 913 01:01:17,280 --> 01:01:19,170 So that's what the initial. 914 01:01:19,170 --> 01:01:24,730 So our money is coming from two sources, this initial deposit, 915 01:01:24,730 --> 01:01:31,800 which was easy, and this continuous, over time deposit, 916 01:01:31,800 --> 01:01:33,730 q of t. 917 01:01:33,730 --> 01:01:37,510 And I have to ask you about that. 918 01:01:37,510 --> 01:01:41,420 That's going to be like the particular solution, 919 01:01:41,420 --> 01:01:45,470 the particular solution that comes from the source term. 920 01:01:45,470 --> 01:01:50,370 This is the solution it comes from the initial condition. 921 01:01:50,370 --> 01:01:52,420 So what do you think this thing looks like? 922 01:01:52,420 --> 01:01:57,470 I just think once we see it, we can say, yeah, 923 01:01:57,470 --> 01:01:58,290 that makes sense. 924 01:02:01,880 --> 01:02:05,070 So now I'm saying what? 925 01:02:05,070 --> 01:02:09,780 If we've deposited q of t in varying amounts, 926 01:02:09,780 --> 01:02:12,950 maybe a constant for a while, maybe a ramp for awhile, 927 01:02:12,950 --> 01:02:23,310 maybe whatever, a step, how am I going to think about this? 928 01:02:23,310 --> 01:02:27,430 So at every time t equal to s, so I'm 929 01:02:27,430 --> 01:02:32,490 using little t for the time I've reached. 930 01:02:32,490 --> 01:02:33,660 Right? 931 01:02:33,660 --> 01:02:37,120 Here's t starting at 0. 932 01:02:37,120 --> 01:02:42,930 Now, let me use s for a time part way along. 933 01:02:42,930 --> 01:02:47,000 So part way along, I input. 934 01:02:47,000 --> 01:02:50,565 I deposit q of s. 935 01:02:50,565 --> 01:02:54,370 I deposit it at time s. 936 01:02:54,370 --> 01:02:56,380 And then what does it do? 937 01:02:56,380 --> 01:02:59,200 That money is in the bank with everybody else. 938 01:02:59,200 --> 01:03:02,170 It grows along with everything else. 939 01:03:02,170 --> 01:03:05,290 So what's the growth factor? 940 01:03:05,290 --> 01:03:07,190 What's the growth factor? 941 01:03:07,190 --> 01:03:11,250 This is the amount I deposited at time s. 942 01:03:11,250 --> 01:03:15,510 And how much has it grown at time t? 943 01:03:15,510 --> 01:03:20,890 This is the key question, and you can answer it. 944 01:03:20,890 --> 01:03:22,380 It went in a time s. 945 01:03:22,380 --> 01:03:23,460 I'm looking at time t. 946 01:03:23,460 --> 01:03:24,210 What's the factor? 947 01:03:24,210 --> 01:03:26,540 AUDIENCE: Is it e to the a t minus s. 948 01:03:26,540 --> 01:03:28,670 PROFESSOR: e to the a t minus s. 949 01:03:39,090 --> 01:03:45,730 So that's the contribution to our balance at time t 950 01:03:45,730 --> 01:03:48,370 from our input at time s. 951 01:03:48,370 --> 01:03:51,800 But now, I've been inputting all the way along. 952 01:03:51,800 --> 01:03:55,380 s is running all the way from here to here. 953 01:03:55,380 --> 01:03:57,660 So finish my formula. 954 01:03:57,660 --> 01:04:00,082 Put me out of my misery. 955 01:04:00,082 --> 01:04:01,690 Or it's not misery, actually. 956 01:04:01,690 --> 01:04:05,020 Its success at this moment. 957 01:04:05,020 --> 01:04:08,470 What do I do now? 958 01:04:08,470 --> 01:04:09,095 I? 959 01:04:09,095 --> 01:04:09,928 AUDIENCE: Integrate. 960 01:04:09,928 --> 01:04:11,360 PROFESSOR: I integrate, exactly. 961 01:04:11,360 --> 01:04:12,720 I integrate. 962 01:04:12,720 --> 01:04:13,970 I integrate. 963 01:04:13,970 --> 01:04:16,600 So all these deposits went in. 964 01:04:16,600 --> 01:04:19,670 They grew that amount in the remaining time. 965 01:04:19,670 --> 01:04:26,680 And I integrate from 0 up to the current time t. 966 01:04:26,680 --> 01:04:29,340 So you see that formula? 967 01:04:29,340 --> 01:04:30,260 Have a look at it. 968 01:04:34,490 --> 01:04:37,800 This is a general formula, and every one of those examples 969 01:04:37,800 --> 01:04:39,830 could be found from that formula. 970 01:04:43,180 --> 01:04:47,690 If q of s was 1, that was our very first example. 971 01:04:47,690 --> 01:04:50,270 We could do that integration. 972 01:04:50,270 --> 01:04:57,170 If q of s was e to the-- anyway, we could do every one. 973 01:04:57,170 --> 01:05:02,350 I just want you to see that that formula makes sense. 974 01:05:02,350 --> 01:05:06,520 Again, this is what grew out of the initial condition. 975 01:05:06,520 --> 01:05:10,700 This is what grew out of the deposit at time s. 976 01:05:10,700 --> 01:05:13,510 And the whole point of calculus, the whole point 977 01:05:13,510 --> 01:05:17,880 of learning [? 1801 ?], the integral equation 978 01:05:17,880 --> 01:05:23,720 part, the integrals part, is integrals just add up. 979 01:05:23,720 --> 01:05:28,950 This term just adds up all the later deposits, 980 01:05:28,950 --> 01:05:34,280 times the growth factor in the remaining time. 981 01:05:34,280 --> 01:05:40,720 And as I say, if I took q of s equal 1-- the examples I gave 982 01:05:40,720 --> 01:05:43,570 are really the examples where you can do the integral. 983 01:05:46,240 --> 01:05:50,830 If q of s is e to the i omega s, I can do that integral. 984 01:05:50,830 --> 01:05:56,390 Actually, it's not hard to do because e to the at doesn't 985 01:05:56,390 --> 01:05:57,210 depend on s. 986 01:05:57,210 --> 01:06:01,980 I can bring an e to the at out in this case. 987 01:06:01,980 --> 01:06:03,870 That formula is just worth thinking about. 988 01:06:03,870 --> 01:06:05,400 It's worth understanding. 989 01:06:05,400 --> 01:06:08,690 I didn't, like, derive it. 990 01:06:08,690 --> 01:06:11,580 And the book does, of course. 991 01:06:11,580 --> 01:06:13,575 There's something called an integrating factor. 992 01:06:16,650 --> 01:06:19,200 You can get at this formula systematically. 993 01:06:19,200 --> 01:06:25,460 I'd rather get at it and understand it. 994 01:06:25,460 --> 01:06:27,800 I'm more interested in understanding 995 01:06:27,800 --> 01:06:32,380 what the meaning of that formula is than the algebra. 996 01:06:32,380 --> 01:06:35,910 Algebra is just a goal to understand, 997 01:06:35,910 --> 01:06:39,020 and that's what I shot for directly. 998 01:06:39,020 --> 01:06:41,750 And as I say, that the book also, 999 01:06:41,750 --> 01:06:46,490 early section of the book, uses practice in calculus. 1000 01:06:46,490 --> 01:06:49,930 Substitute that in to the equation. 1001 01:06:49,930 --> 01:06:52,120 Figure out what is dy dt. 1002 01:06:52,120 --> 01:06:56,470 And check that it works. 1003 01:06:56,470 --> 01:07:05,080 It's worth actually looking at that end of what 1004 01:07:05,080 --> 01:07:07,290 you need to know from calculus It's is. 1005 01:07:07,290 --> 01:07:10,530 You should be able to plug that in for y 1006 01:07:10,530 --> 01:07:14,090 and see that solves the equation. 1007 01:07:14,090 --> 01:07:22,930 Right, now I have enough time to do cosine omega t. 1008 01:07:22,930 --> 01:07:27,710 But I don't have enough time to do it the complex way. 1009 01:07:27,710 --> 01:07:36,280 So let me do as a final example, the equation. 1010 01:07:36,280 --> 01:07:37,120 Let me just think. 1011 01:07:37,120 --> 01:07:39,630 I don't know if I have enough space here. 1012 01:07:39,630 --> 01:07:43,960 I'm now going to do dy dt-- can I call that y prime to save 1013 01:07:43,960 --> 01:07:53,160 a little space-- equal ay plus cosine of t. 1014 01:07:53,160 --> 01:07:54,721 I'll take omega to be 1. 1015 01:07:58,650 --> 01:08:02,682 Now, how could we solve that one? 1016 01:08:05,410 --> 01:08:09,690 I'm going to solve it without complex numbers, 1017 01:08:09,690 --> 01:08:12,960 just to see how easy or hard that is. 1018 01:08:12,960 --> 01:08:17,130 And you'll see, actually, it's easy. 1019 01:08:17,130 --> 01:08:20,050 But complex numbers will tell us more. 1020 01:08:20,050 --> 01:08:25,279 So it's easy, but not totally easy. 1021 01:08:25,279 --> 01:08:29,010 So what did I do in the earlier example 1022 01:08:29,010 --> 01:08:32,970 if the right hand side was a 1, a constant? 1023 01:08:32,970 --> 01:08:35,500 I look for the solution to be a constant. 1024 01:08:35,500 --> 01:08:38,240 If the right hand side was an exponential, 1025 01:08:38,240 --> 01:08:41,010 I look for the solution to be an exponential. 1026 01:08:41,010 --> 01:08:46,180 Now, my right hand side, my source term, is a cosine. 1027 01:08:46,180 --> 01:08:49,479 So what form of the solution am I going to look for? 1028 01:08:52,020 --> 01:08:54,329 I naturally think, OK, look for a cosine. 1029 01:08:57,859 --> 01:09:04,090 We could try y equals some number M cosine t. 1030 01:09:07,899 --> 01:09:14,109 Now, you have to see what goes wrong and how to fix it. 1031 01:09:14,109 --> 01:09:17,640 So if I plug that in, looking for M the same way 1032 01:09:17,640 --> 01:09:20,684 I look for capital Y earlier, I plug this in, 1033 01:09:20,684 --> 01:09:24,439 and I get aM cosine t cosine t. 1034 01:09:24,439 --> 01:09:27,240 But what do I get for y prime? 1035 01:09:27,240 --> 01:09:28,020 Sine t. 1036 01:09:30,910 --> 01:09:32,160 And I can't match. 1037 01:09:32,160 --> 01:09:33,979 I can make it work. 1038 01:09:33,979 --> 01:09:38,050 I can't make a sine there magic a cosine here. 1039 01:09:38,050 --> 01:09:40,510 So what's the solution? 1040 01:09:40,510 --> 01:09:43,620 How do I fix it? 1041 01:09:43,620 --> 01:09:47,529 I better allow my solution to include 1042 01:09:47,529 --> 01:09:51,563 some sine plus N sine t. 1043 01:09:56,110 --> 01:10:02,240 So that's the problem with doing it, keeping things real. 1044 01:10:02,240 --> 01:10:06,750 I'll push this through, no problem. 1045 01:10:06,750 --> 01:10:09,180 But cosine by itself won't work. 1046 01:10:09,180 --> 01:10:13,820 I need to have sines there, because derivatives bring out 1047 01:10:13,820 --> 01:10:14,700 sines. 1048 01:10:14,700 --> 01:10:18,570 So I have a combination of cosine and sine. 1049 01:10:18,570 --> 01:10:20,933 I have a combination of cosine and sine. 1050 01:10:24,140 --> 01:10:27,140 So the complex method will work in one shot 1051 01:10:27,140 --> 01:10:36,410 because e to the i omega t is a combination of cosine and sine. 1052 01:10:36,410 --> 01:10:39,560 Or another way to say it is when I see cosine here, 1053 01:10:39,560 --> 01:10:41,410 that's got two exponentials. 1054 01:10:41,410 --> 01:10:45,030 That's got e to the it and e to the-- anyway. 1055 01:10:45,030 --> 01:10:47,180 Let's go for the real one. 1056 01:10:47,180 --> 01:10:48,790 So I'm going to plug that into there. 1057 01:10:52,860 --> 01:10:56,560 So I'll get sines and cosines, right? 1058 01:10:56,560 --> 01:10:59,180 When I plug this into there, I'll 1059 01:10:59,180 --> 01:11:01,480 have some sines and some cosines, 1060 01:11:01,480 --> 01:11:03,980 and I'll just match the two separately. 1061 01:11:03,980 --> 01:11:06,190 So I'm going to get two equations. 1062 01:11:06,190 --> 01:11:11,370 First of all, let me say what's the cosine equation? 1063 01:11:11,370 --> 01:11:14,130 And then what's the sine equation? 1064 01:11:14,130 --> 01:11:18,190 So when I match cosine terms, what do I have? 1065 01:11:18,190 --> 01:11:21,300 What cosine terms do I get out of y prime, here? 1066 01:11:23,970 --> 01:11:26,290 The derivative. 1067 01:11:26,290 --> 01:11:28,470 Well, the derivative of cosine is a sine. 1068 01:11:28,470 --> 01:11:30,560 That that's not a cosine term. 1069 01:11:30,560 --> 01:11:32,780 The derivative of sine is cosine. 1070 01:11:32,780 --> 01:11:38,000 I think I get, if I just match cosines, 1071 01:11:38,000 --> 01:11:39,740 I think I get an N cosine. 1072 01:11:39,740 --> 01:11:44,940 N cosine t equal ay. 1073 01:11:44,940 --> 01:11:49,400 How many cosines do I have from that term? 1074 01:11:49,400 --> 01:11:52,040 ay has an M cosine t. 1075 01:11:52,040 --> 01:11:55,832 I think I have an aM, and here I've got 1. 1076 01:11:59,990 --> 01:12:07,070 That was a natural step, but new to us. 1077 01:12:07,070 --> 01:12:09,940 I'm matching the cosines. 1078 01:12:09,940 --> 01:12:15,440 I have on the left side, with this form of the solution, 1079 01:12:15,440 --> 01:12:18,020 the derivative will have an N cosine t. 1080 01:12:18,020 --> 01:12:22,850 So I had N cosines, aM cosines, and 1 cosine. 1081 01:12:22,850 --> 01:12:24,760 Now, what if I match signs? 1082 01:12:24,760 --> 01:12:26,500 What happens there? 1083 01:12:26,500 --> 01:12:29,900 We're pushing more than an hour, so 1084 01:12:29,900 --> 01:12:34,020 hang on for another five minutes, and we're there. 1085 01:12:34,020 --> 01:12:38,310 Now, what happens if I match sines, sine t? 1086 01:12:38,310 --> 01:12:40,510 How do I get sine t in y prime? 1087 01:12:44,740 --> 01:12:47,734 So take the derivative of that, and what do you have? 1088 01:12:47,734 --> 01:12:48,900 AUDIENCE: Minus [INAUDIBLE]. 1089 01:12:48,900 --> 01:12:51,190 PROFESSOR: Minus M sine t. 1090 01:12:51,190 --> 01:12:55,170 That tells me how many sine t's are in there. 1091 01:12:55,170 --> 01:12:58,690 And on the right hand side, a times y, 1092 01:12:58,690 --> 01:13:01,379 how many sine t's do I have from that? 1093 01:13:01,379 --> 01:13:02,420 AUDIENCE: You have N t's. 1094 01:13:02,420 --> 01:13:04,740 PROFESSOR: N, good thinking. 1095 01:13:04,740 --> 01:13:09,210 And what about from this term? 1096 01:13:09,210 --> 01:13:11,800 None, no sine there. 1097 01:13:11,800 --> 01:13:18,940 So I have two equations by matching the cosines and sines. 1098 01:13:18,940 --> 01:13:22,360 Once you see it, you could do it again. 1099 01:13:22,360 --> 01:13:25,240 And we can solve those equations, 1100 01:13:25,240 --> 01:13:30,290 two ordinary, very simple equations for M and N. 1101 01:13:30,290 --> 01:13:32,180 Let's see if I make space. 1102 01:13:32,180 --> 01:13:38,210 Why don't I do it here, so you can see it. 1103 01:13:41,260 --> 01:13:43,780 So how do I solve those two equations? 1104 01:13:43,780 --> 01:13:50,070 Well, this equation gives me-- easy-- gives me M as minus aN. 1105 01:13:50,070 --> 01:13:51,930 So I'll just put that in for N. 1106 01:13:51,930 --> 01:13:58,086 So I have N equals aM. 1107 01:13:58,086 --> 01:14:00,360 But M is minus aN. 1108 01:14:00,360 --> 01:14:06,264 I think I've got minus a squared N plus that 1. 1109 01:14:11,990 --> 01:14:17,570 All I did was solve the equation, just by common sense. 1110 01:14:17,570 --> 01:14:21,860 You could say by linear algebra, but linear algebra's 1111 01:14:21,860 --> 01:14:24,310 got a little more to it than this. 1112 01:14:24,310 --> 01:14:29,720 So now I know M, and now I know N. So now I know the answer. 1113 01:14:29,720 --> 01:14:38,944 y is M, so M is minus aN. 1114 01:14:41,590 --> 01:14:44,690 Oh, well, I have to figure out what N is, here. 1115 01:14:44,690 --> 01:14:46,510 What is N? 1116 01:14:46,510 --> 01:14:50,570 This is giving me N, but I better figure it out. 1117 01:14:50,570 --> 01:14:53,280 What is N from that first equation? 1118 01:14:53,280 --> 01:14:56,000 And then I'll plug in. 1119 01:14:56,000 --> 01:14:57,843 And then I'm quit. 1120 01:14:57,843 --> 01:14:58,769 AUDIENCE: [INAUDIBLE]. 1121 01:14:58,769 --> 01:15:01,084 PROFESSOR: 1 over, yeah. 1122 01:15:01,084 --> 01:15:02,480 AUDIENCE: 1 plus a squared. 1123 01:15:02,480 --> 01:15:05,530 PROFESSOR: 1 plus a squared, good. 1124 01:15:05,530 --> 01:15:07,530 Because that term goes over there, and we have 1 1125 01:15:07,530 --> 01:15:08,650 plus a squared. 1126 01:15:08,650 --> 01:15:13,980 So now y is M cosine t. 1127 01:15:13,980 --> 01:15:16,110 So M is minus aN. 1128 01:15:16,110 --> 01:15:24,990 So minus aN is 1 over 1 plus a squared cosine t. 1129 01:15:24,990 --> 01:15:26,580 Is that right? 1130 01:15:26,580 --> 01:15:28,940 That was the cosines. 1131 01:15:28,940 --> 01:15:35,330 And we had N sine t. 1132 01:15:35,330 --> 01:15:38,710 But N is just 1-- I think I just add the sine t. 1133 01:15:38,710 --> 01:15:40,284 Have I got it? 1134 01:15:40,284 --> 01:15:41,760 I think so. 1135 01:15:46,190 --> 01:15:51,520 Here is the N sine t, and here is the M cos t. 1136 01:15:55,365 --> 01:15:56,415 It was just algebra. 1137 01:16:01,054 --> 01:16:02,470 Typical of these problems, there's 1138 01:16:02,470 --> 01:16:05,970 a little thinking and then some algebra. 1139 01:16:05,970 --> 01:16:10,350 The thinking led us to this. 1140 01:16:10,350 --> 01:16:14,420 The thinking led us to the fact we needed cosines in there, 1141 01:16:14,420 --> 01:16:16,650 as well as cosines. 1142 01:16:16,650 --> 01:16:18,590 But then once we did it, then the thinking 1143 01:16:18,590 --> 01:16:23,275 said, OK, separately match the cosine terms and the sine term. 1144 01:16:23,275 --> 01:16:24,690 And then do the algebra. 1145 01:16:27,260 --> 01:16:38,230 Now, I just want to do this with complex. 1146 01:16:38,230 --> 01:16:44,260 So y prime equals ay plus e to the it. 1147 01:16:49,464 --> 01:16:51,660 To get an idea, you see the two. 1148 01:16:51,660 --> 01:16:54,000 And then I have to talk about it. 1149 01:16:54,000 --> 01:16:56,820 You see, I'm only going to go part way with this 1150 01:16:56,820 --> 01:17:00,810 and then save it for Wednesday. 1151 01:17:00,810 --> 01:17:06,470 But if I see this, what solution do I assume? 1152 01:17:06,470 --> 01:17:16,304 This is like an e to the st. I assume y is some Y e to the it. 1153 01:17:16,304 --> 01:17:19,340 See, I don't have cosines and sines anymore. 1154 01:17:19,340 --> 01:17:20,680 I have e to the it. 1155 01:17:20,680 --> 01:17:22,960 And if I take the derivative of e to the it, 1156 01:17:22,960 --> 01:17:25,180 I'm still in the e to the it world. 1157 01:17:25,180 --> 01:17:25,870 So I do this. 1158 01:17:28,710 --> 01:17:29,790 I plug it in. 1159 01:17:29,790 --> 01:17:32,730 Uh-huh, let me leave that for Wednesday. 1160 01:17:32,730 --> 01:17:36,390 We have to have some excitement for Wednesday. 1161 01:17:36,390 --> 01:17:41,220 So we'll get a complex answer, and then we'll 1162 01:17:41,220 --> 01:17:43,570 take the real part to solve that problem. 1163 01:17:46,370 --> 01:17:49,880 So we've got two steps, one way or the other way. 1164 01:17:49,880 --> 01:17:54,300 Here, we had two steps because we had to let sines sneak in. 1165 01:17:54,300 --> 01:17:58,920 Here, we have two steps because I could solve it, 1166 01:17:58,920 --> 01:18:02,160 and you could solve that right away. 1167 01:18:02,160 --> 01:18:05,370 But then you have to take the real part. 1168 01:18:05,370 --> 01:18:06,990 I'll leave that. 1169 01:18:06,990 --> 01:18:08,905 Is there questions? 1170 01:18:08,905 --> 01:18:12,606 Do you want me to recap quickly what we've done. 1171 01:18:12,606 --> 01:18:15,450 AUDIENCE: Yes. 1172 01:18:15,450 --> 01:18:17,510 PROFESSOR: I try to leave on the board 1173 01:18:17,510 --> 01:18:21,500 enough to make a recap possible. 1174 01:18:21,500 --> 01:18:25,020 Everything was about that equation. 1175 01:18:25,020 --> 01:18:28,480 We have only solved-- I shouldn't 1176 01:18:28,480 --> 01:18:33,490 say only-- we have solved the constant coefficient, model 1177 01:18:33,490 --> 01:18:35,770 constant coefficient, first order equation. 1178 01:18:42,480 --> 01:18:44,165 Wednesday comes nonlinear equation. 1179 01:18:46,760 --> 01:18:49,330 This one today was strictly linear. 1180 01:18:49,330 --> 01:18:51,470 So what did we do? 1181 01:18:51,470 --> 01:18:57,670 We solved this equation, first of all, for q equal 1; 1182 01:18:57,670 --> 01:19:02,220 secondly, for q equal e to the st; thirdly, 1183 01:19:02,220 --> 01:19:13,680 for q equal a step; fourthly for q equal-- where is it? 1184 01:19:13,680 --> 01:19:14,790 Where is that delta of t? 1185 01:19:14,790 --> 01:19:17,430 Maybe it's here. 1186 01:19:17,430 --> 01:19:19,450 Ah, it got erased. 1187 01:19:19,450 --> 01:19:28,610 So the fourth guy was y prime equal ay plus delta of t, 1188 01:19:28,610 --> 01:19:34,730 or delta of t minus capital T. So those 1189 01:19:34,730 --> 01:19:36,010 were our four examples. 1190 01:19:38,850 --> 01:19:41,280 And then what did we finally do? 1191 01:19:44,500 --> 01:19:48,200 So if we're recapping, compressing, 1192 01:19:48,200 --> 01:19:51,170 we're compressing everything into two minutes. 1193 01:19:51,170 --> 01:19:57,010 We solved those four examples, and then we 1194 01:19:57,010 --> 01:20:00,100 solved the general problem. 1195 01:20:00,100 --> 01:20:01,810 And when we solved the general problem, 1196 01:20:01,810 --> 01:20:06,920 that gave us this integral, which my whole goal was 1197 01:20:06,920 --> 01:20:11,120 that you should understand that this should seem right to you. 1198 01:20:11,120 --> 01:20:15,000 This is adding up the value at time t 1199 01:20:15,000 --> 01:20:20,350 from all the inputs at different times s. 1200 01:20:20,350 --> 01:20:24,850 So to add them up, we integrate from 0 to t. 1201 01:20:24,850 --> 01:20:27,880 And finally, we returned to the question 1202 01:20:27,880 --> 01:20:31,000 of cos t, all important question. 1203 01:20:31,000 --> 01:20:33,990 But awkward question, because we needed to let sine 1204 01:20:33,990 --> 01:20:36,140 t in there too.