1 00:00:06,700 --> 00:00:11,170 GILBERT STRANG: OK, today is about differential equations. 2 00:00:11,170 --> 00:00:14,730 That's where calculus really is applied. 3 00:00:14,730 --> 00:00:18,490 And these will be equations that describe growth. 4 00:00:18,490 --> 00:00:21,540 And the first you've already seen. 5 00:00:21,540 --> 00:00:25,680 It's the most important and the simplest. The growth rate 6 00:00:25,680 --> 00:00:31,190 dy dt is proportional to y itself. 7 00:00:31,190 --> 00:00:35,550 Let's call c that constant that comes in there. 8 00:00:35,550 --> 00:00:41,560 And all these problems will start from some known point y 9 00:00:41,560 --> 00:00:48,890 at time 0 and evolve, grow, to k, whatever they do. 10 00:00:48,890 --> 00:00:52,410 And this one we know the solution to. 11 00:00:52,410 --> 00:00:58,410 We know that that will be an exponential growth with a 12 00:00:58,410 --> 00:01:01,220 factor c, a growth rate c. 13 00:01:01,220 --> 00:01:06,990 Actually, we should know that the solution has an e to the 14 00:01:06,990 --> 00:01:14,190 ct because when we take the derivative of this, it will 15 00:01:14,190 --> 00:01:17,550 bring down that factor c that we want. 16 00:01:17,550 --> 00:01:22,820 And at t equals 0 this is correct. 17 00:01:22,820 --> 00:01:26,180 We get started correctly because at t equals 0 we have 18 00:01:26,180 --> 00:01:30,940 y of 0 is correctly given by the known starting point. 19 00:01:30,940 --> 00:01:34,020 Because when t is 0, that's 1. 20 00:01:34,020 --> 00:01:35,990 So that's the one we know. 21 00:01:35,990 --> 00:01:37,350 The most fundamental. 22 00:01:37,350 --> 00:01:40,830 Now, the next step. 23 00:01:40,830 --> 00:01:43,610 Still linear. 24 00:01:43,610 --> 00:01:48,280 That allows a source term. 25 00:01:48,280 --> 00:01:54,200 This might be money in the bank growing with an interest 26 00:01:54,200 --> 00:01:57,020 rate c per year. 27 00:01:57,020 --> 00:02:00,700 And the source term would be additional money that you're 28 00:02:00,700 --> 00:02:02,525 constantly putting in. 29 00:02:05,370 --> 00:02:12,220 Every time step in goes that saving. 30 00:02:12,220 --> 00:02:16,920 Or if s were negative, of course, it could be spending. 31 00:02:16,920 --> 00:02:23,800 For us, that's a linear differential equation with a 32 00:02:23,800 --> 00:02:27,780 constant right-hand side there. 33 00:02:27,780 --> 00:02:30,720 And we have to be able to solve it. 34 00:02:30,720 --> 00:02:33,730 And we can. 35 00:02:33,730 --> 00:02:38,490 In fact, I can fit it into the one we already now. 36 00:02:38,490 --> 00:02:40,070 Watch this. 37 00:02:40,070 --> 00:02:44,805 Let me take that to be c times y plus s/c. 38 00:02:48,050 --> 00:02:51,490 Remember, s and c are constants. 39 00:02:51,490 --> 00:02:58,060 And let me write that right-hand side, dy dt. 40 00:02:58,060 --> 00:03:01,450 I can put this constant in there too and still have the 41 00:03:01,450 --> 00:03:02,300 same derivative. 42 00:03:02,300 --> 00:03:08,600 So dy dt is exactly the same as the derivative of y plus 43 00:03:08,600 --> 00:03:15,190 s/c because that s/c is constant; its derivative is 0. 44 00:03:15,190 --> 00:03:17,490 So that dy dt is the same as before. 45 00:03:17,490 --> 00:03:19,520 But now look. 46 00:03:19,520 --> 00:03:24,530 We have this expression here that's just like this one, 47 00:03:24,530 --> 00:03:32,100 only it's y plus s/c that's growing at this growth rate c, 48 00:03:32,100 --> 00:03:35,040 starting from-- 49 00:03:35,040 --> 00:03:40,800 the start would be y at 0 plus s/c. 50 00:03:40,800 --> 00:03:45,360 You see, this quantity in parentheses, I'm grabbing that 51 00:03:45,360 --> 00:03:49,080 as the thing that grows perfect exponentially. 52 00:03:49,080 --> 00:03:58,580 So I conclude that at a later time its value is its initial 53 00:03:58,580 --> 00:04:03,930 value times the growth. 54 00:04:06,740 --> 00:04:12,200 That is a rather quick solution to this 55 00:04:12,200 --> 00:04:14,210 differential equation. 56 00:04:14,210 --> 00:04:19,279 You might want me to put s/c on the right-hand side and 57 00:04:19,279 --> 00:04:23,130 have y of t by itself, and that would be a formula for 58 00:04:23,130 --> 00:04:27,260 the correct y of t that solves this equation that 59 00:04:27,260 --> 00:04:30,360 starts at y of 0. 60 00:04:30,360 --> 00:04:34,870 And of course, this one starts at y of 0 plus s/c and that's 61 00:04:34,870 --> 00:04:39,605 why we see y of 0 plus s/c right there. 62 00:04:39,605 --> 00:04:44,990 Do you see that that equation, well, it wasn't really 63 00:04:44,990 --> 00:04:46,010 systematic. 64 00:04:46,010 --> 00:04:50,150 And in a differential equations course you would 65 00:04:50,150 --> 00:04:54,890 learn how to reach that answer without 66 00:04:54,890 --> 00:04:57,250 sort of noticing that-- 67 00:04:57,250 --> 00:05:01,080 actually, I should do that too. 68 00:05:01,080 --> 00:05:04,985 This is such a useful, important equation. 69 00:05:04,985 --> 00:05:10,520 It's linear, but it's got some right-hand side there. 70 00:05:10,520 --> 00:05:16,320 I should give you a system, say something about linear 71 00:05:16,320 --> 00:05:17,990 equations before I go on. 72 00:05:21,720 --> 00:05:26,630 The most interesting equation will be not linear, but let me 73 00:05:26,630 --> 00:05:29,410 say a word about linear equations of which that's a 74 00:05:29,410 --> 00:05:32,290 perfect example with a right-hand side. 75 00:05:32,290 --> 00:05:33,540 So linear equations. 76 00:05:36,320 --> 00:05:37,570 Linear equations. 77 00:05:41,370 --> 00:05:49,590 The solutions to linear equations, y of t, is-- 78 00:05:49,590 --> 00:05:50,820 to all linear equations. 79 00:05:50,820 --> 00:05:54,280 Actually, this is a linear differential equation. 80 00:05:54,280 --> 00:05:56,930 When I'm teaching linear algebra I'm talking about 81 00:05:56,930 --> 00:05:58,180 matrix equations. 82 00:06:00,620 --> 00:06:08,730 The rule, it's the linear part that is important for these 83 00:06:08,730 --> 00:06:10,190 few thoughts. 84 00:06:10,190 --> 00:06:13,200 The solution to linear equations-- 85 00:06:13,200 --> 00:06:14,595 can I underline linear-- 86 00:06:17,710 --> 00:06:23,335 always has the form of some particular solution. 87 00:06:26,990 --> 00:06:36,770 Some function that does solve the equation plus another 88 00:06:36,770 --> 00:06:40,270 solution with a right side 0. 89 00:06:49,350 --> 00:06:53,050 You will see what I mean by those two 90 00:06:53,050 --> 00:06:55,350 parts for this example. 91 00:06:55,350 --> 00:07:04,350 Can I copy again this example dy dt is cy plus s. 92 00:07:04,350 --> 00:07:07,150 So I'm looking for a solution to that equation. 93 00:07:07,150 --> 00:07:10,240 First, let me look for a particular solution. 94 00:07:10,240 --> 00:07:15,730 That means just any function that solves the equation. 95 00:07:15,730 --> 00:07:18,680 Well, the simplest function I could think of would be a 96 00:07:18,680 --> 00:07:21,001 constant function. 97 00:07:21,001 --> 00:07:26,210 And if my solution was a constant, then its derivative 98 00:07:26,210 --> 00:07:26,950 would be 0. 99 00:07:26,950 --> 00:07:28,620 So that would be 0. 100 00:07:28,620 --> 00:07:32,205 And then c times that constant plus s would be 0. 101 00:07:34,940 --> 00:07:38,520 I want the equation to be solved. 102 00:07:38,520 --> 00:07:43,990 I claim that a particular solution is the constant y-- 103 00:07:43,990 --> 00:07:48,445 so when I set that momentarily to 0, I'll discover I'll move 104 00:07:48,445 --> 00:07:50,640 the s over as minus s. 105 00:07:50,640 --> 00:07:54,550 I'll divide by the c and I'll have a minus s/c. 106 00:07:57,180 --> 00:07:58,750 That's a particular solution. 107 00:07:58,750 --> 00:08:02,180 Let's just plug it in. 108 00:08:02,180 --> 00:08:03,300 If I put in y. 109 00:08:03,300 --> 00:08:06,340 That's a constant, so it's derivative is 0. 110 00:08:06,340 --> 00:08:11,410 And c times that gives me minus s plus s is the 0. 111 00:08:11,410 --> 00:08:12,660 OK. 112 00:08:16,850 --> 00:08:19,060 That's a particular solution. 113 00:08:19,060 --> 00:08:22,500 And then what I mean by right side 0. 114 00:08:22,500 --> 00:08:24,160 What do I mean by that? 115 00:08:24,160 --> 00:08:27,490 I'm not going to wipe out all this, I'm going to wipe out-- 116 00:08:27,490 --> 00:08:33,679 for this right side 0 part, I'm going to wipe out the 117 00:08:33,679 --> 00:08:35,679 source term. 118 00:08:35,679 --> 00:08:39,860 I mean, I keep the y's, but throw away the source in the 119 00:08:39,860 --> 00:08:40,760 right side 0. 120 00:08:40,760 --> 00:08:46,180 Maybe a book might often write the homogeneous equation. 121 00:08:48,700 --> 00:08:51,270 And the homogeneous equation is where we started, 122 00:08:51,270 --> 00:08:52,645 dy dt equals cy. 123 00:08:56,780 --> 00:08:59,040 What's the solution to that? 124 00:08:59,040 --> 00:09:01,330 The solution to dy dt is cy. 125 00:09:04,160 --> 00:09:06,460 This is this solution, e to the ct. 126 00:09:10,330 --> 00:09:18,210 Any a, any a times e to the ct will solve my simple equation. 127 00:09:18,210 --> 00:09:23,490 Once I've knocked out this s, then if y solves it, a 128 00:09:23,490 --> 00:09:24,850 times y solves it. 129 00:09:24,850 --> 00:09:27,080 I could multiply the equation by a. 130 00:09:27,080 --> 00:09:31,700 So a is arbitrary, any number. 131 00:09:31,700 --> 00:09:36,270 But of course, I'm going to find out what it is by 132 00:09:36,270 --> 00:09:40,260 starting at the right place. 133 00:09:40,260 --> 00:09:43,370 By starting at y of 0, I'll find out what-- 134 00:09:43,370 --> 00:09:45,130 by setting t equals 0. 135 00:09:45,130 --> 00:09:47,560 So put t equals 0. 136 00:09:47,560 --> 00:09:54,390 To find a, put t equals 0. 137 00:09:54,390 --> 00:09:56,350 Because we know where we started. 138 00:09:56,350 --> 00:10:04,760 So we started at a known y of 0 equals minus s/c plus A. 139 00:10:04,760 --> 00:10:08,110 Setting t equals 0 makes that 1. 140 00:10:08,110 --> 00:10:11,100 You see we found out what A is. 141 00:10:11,100 --> 00:10:14,920 A is y of 0 plus s/c. 142 00:10:14,920 --> 00:10:21,280 And that's the thing that grows, that multiplies 143 00:10:21,280 --> 00:10:22,690 the e to the ct. 144 00:10:22,690 --> 00:10:25,620 And of course, it did it up here. 145 00:10:25,620 --> 00:10:31,320 Up here what multiplied the e to the ct was this y of 0 plus 146 00:10:31,320 --> 00:10:40,690 s/c, which is exactly our A. 147 00:10:40,690 --> 00:10:45,400 As I said, my main point in this lecture is to bring in a 148 00:10:45,400 --> 00:10:47,330 nonlinear equation. 149 00:10:47,330 --> 00:10:51,540 That's quite interesting and important. 150 00:10:51,540 --> 00:10:57,780 And it comes in population growth, ecology, it appears a 151 00:10:57,780 --> 00:11:00,570 lot of places and I'll write it down immediately. 152 00:11:00,570 --> 00:11:05,630 But just the starting point is always linear. 153 00:11:05,630 --> 00:11:10,870 And here we've got the basic linear equation and now the 154 00:11:10,870 --> 00:11:16,510 basic linear equation with a source term. 155 00:11:16,510 --> 00:11:20,580 Ready for the population equation. 156 00:11:20,580 --> 00:11:23,220 Population growth. 157 00:11:23,220 --> 00:11:26,980 Population P of t. 158 00:11:26,980 --> 00:11:33,620 OK, what's a reasonable differential equation that 159 00:11:33,620 --> 00:11:36,390 describes the growth of population? 160 00:11:36,390 --> 00:11:41,950 All right, so I'm interested in, what is a model? 161 00:11:41,950 --> 00:11:46,140 So this is actually where mathematics is applied. 162 00:11:46,140 --> 00:11:49,070 And many people will have models. 163 00:11:49,070 --> 00:11:52,290 And people who have a census can check those models and 164 00:11:52,290 --> 00:11:53,730 see, well, what are the constants? 165 00:11:56,830 --> 00:11:59,880 Does a good choice of constant make this model realistic over 166 00:11:59,880 --> 00:12:01,780 the last hundred years? 167 00:12:01,780 --> 00:12:02,780 Lots to do. 168 00:12:02,780 --> 00:12:05,900 Fantastic projects in this area. 169 00:12:05,900 --> 00:12:10,550 If you're looking for a project, google "population 170 00:12:10,550 --> 00:12:14,760 growth." See what they say. 171 00:12:14,760 --> 00:12:15,950 Several sites will say-- 172 00:12:15,950 --> 00:12:17,610 Wikipedia, I did it this morning. 173 00:12:17,610 --> 00:12:18,710 I looked at Wikipedia. 174 00:12:18,710 --> 00:12:22,750 I'll tell you some things later about Wikipedia. 175 00:12:22,750 --> 00:12:27,750 And also, the Census Bureau has a site and they all know 176 00:12:27,750 --> 00:12:31,760 the population now pretty closely. 177 00:12:31,760 --> 00:12:33,470 Everybody would like to know what it's 178 00:12:33,470 --> 00:12:36,170 going to be in 50 years. 179 00:12:36,170 --> 00:12:41,860 And to get 50 years out, you have to have some 180 00:12:41,860 --> 00:12:43,600 mathematical model. 181 00:12:43,600 --> 00:12:47,610 And I'm going to pick this differential equation. 182 00:12:47,610 --> 00:12:48,860 There's going to be a growth. 183 00:12:52,350 --> 00:12:55,670 That c would be sort of birth rate. 184 00:12:55,670 --> 00:13:00,890 Well, I guess birth rate minus death rate. 185 00:13:00,890 --> 00:13:03,140 So that would be the growth rate. 186 00:13:03,140 --> 00:13:08,060 And if we only had that term the population would grow 187 00:13:08,060 --> 00:13:11,530 exponentially forever. 188 00:13:11,530 --> 00:13:13,910 There's going to be another term because that's not 189 00:13:13,910 --> 00:13:18,540 realistic for the population just to keep growing. 190 00:13:18,540 --> 00:13:27,680 As the earth fills up, there's some maybe competition term. 191 00:13:27,680 --> 00:13:30,330 Some slowdown term. 192 00:13:30,330 --> 00:13:32,820 So it's going to have a minus sign. 193 00:13:32,820 --> 00:13:35,990 And maybe s for a slow down factor. 194 00:13:35,990 --> 00:13:37,630 And I'm going to take P squared. 195 00:13:40,470 --> 00:13:44,850 That reflects sort of population interacting with 196 00:13:44,850 --> 00:13:46,390 population. 197 00:13:46,390 --> 00:13:50,890 There are many problems. This is also a model for problems 198 00:13:50,890 --> 00:13:54,420 in chemistry and biology, mass action it would 199 00:13:54,420 --> 00:13:55,930 be called in chemistry. 200 00:13:55,930 --> 00:13:59,570 Where I would have two different materials, two 201 00:13:59,570 --> 00:14:03,170 different substances, two different chemicals. 202 00:14:03,170 --> 00:14:07,420 And the interaction between the two would be proportional 203 00:14:07,420 --> 00:14:11,670 to the amount of one and also proportional to the 204 00:14:11,670 --> 00:14:12,730 amount of the other. 205 00:14:12,730 --> 00:14:17,280 So it'll be proportional to the product. 206 00:14:17,280 --> 00:14:20,630 One concentration times the other. 207 00:14:20,630 --> 00:14:23,230 Well, there I would have to have two equations for the two 208 00:14:23,230 --> 00:14:24,530 concentrations. 209 00:14:24,530 --> 00:14:29,680 Here I'm taking a more basic problem where I just have one 210 00:14:29,680 --> 00:14:33,980 unknown, the population P, and one equation for it. 211 00:14:33,980 --> 00:14:38,860 And it's P times P. It's the number of people meeting other 212 00:14:38,860 --> 00:14:42,840 people and crowding. 213 00:14:42,840 --> 00:14:45,130 So s is a very small number. 214 00:14:45,130 --> 00:14:48,255 Very small, like one over a billion or something. 215 00:14:53,180 --> 00:14:54,960 Now, I want you to solve that equation. 216 00:14:57,770 --> 00:15:00,250 Or I want us to solve it. 217 00:15:00,250 --> 00:15:04,510 And we don't right now have great tools 218 00:15:04,510 --> 00:15:06,860 for nonlinear equations. 219 00:15:06,860 --> 00:15:11,390 We have more calculus to learn, but this one will give 220 00:15:11,390 --> 00:15:13,210 in if we do it right. 221 00:15:13,210 --> 00:15:16,370 And let me show you what the solution looks like because 222 00:15:16,370 --> 00:15:18,730 you don't want to miss that. 223 00:15:18,730 --> 00:15:23,010 We'll get into the formula. 224 00:15:23,010 --> 00:15:25,790 We'll make it as nice as possible, but 225 00:15:25,790 --> 00:15:27,600 the graph is great. 226 00:15:27,600 --> 00:15:29,280 So the population. 227 00:15:29,280 --> 00:15:31,950 Well, notice, what would happen if the population 228 00:15:31,950 --> 00:15:33,810 starts out at 0? 229 00:15:33,810 --> 00:15:36,650 Nobody's around. 230 00:15:36,650 --> 00:15:39,590 Then the derivative is 0. 231 00:15:39,590 --> 00:15:42,270 It never leaves 0. 232 00:15:42,270 --> 00:15:46,460 So this 0 would be a solution. 233 00:15:46,460 --> 00:15:48,920 P equals 0 constant solution. 234 00:15:48,920 --> 00:15:50,090 Not interesting. 235 00:15:50,090 --> 00:15:54,420 There's another case, important case, when the 236 00:15:54,420 --> 00:15:57,010 derivative is 0. 237 00:15:57,010 --> 00:15:58,640 Suppose the derivative is 0. 238 00:15:58,640 --> 00:16:01,770 That means we're going to look for a particular very special 239 00:16:01,770 --> 00:16:05,690 solution that doesn't move. 240 00:16:05,690 --> 00:16:11,080 So if that derivative is 0, then cP equals sP squared. 241 00:16:11,080 --> 00:16:13,220 Can I do this in the corner? 242 00:16:13,220 --> 00:16:16,720 If cP equals sP squared. 243 00:16:16,720 --> 00:16:23,680 And if I cancel a P, I get P is equal to c/s. 244 00:16:23,680 --> 00:16:25,660 So there's some solution. 245 00:16:25,660 --> 00:16:27,340 Can I call it that c/s? 246 00:16:30,090 --> 00:16:33,220 Well, everybody would like to know exactly what those 247 00:16:33,220 --> 00:16:35,190 numbers are. 248 00:16:35,190 --> 00:16:39,860 It's maybe about 10 billion people. 249 00:16:39,860 --> 00:16:42,840 So what I'm saying is if I got up to-- 250 00:16:42,840 --> 00:16:43,650 not me. 251 00:16:43,650 --> 00:16:49,120 If the world got up to 10 billion, then at that point, 252 00:16:49,120 --> 00:16:53,490 the growth and the competition would knock each other out, 253 00:16:53,490 --> 00:16:55,310 would cancel each other. 254 00:16:55,310 --> 00:16:57,540 There wouldn't be any further change. 255 00:16:57,540 --> 00:17:02,780 So that's a kind of point that we won't go past. 256 00:17:02,780 --> 00:17:06,650 But now, let me draw the real solution because the real 257 00:17:06,650 --> 00:17:08,940 solution starts with-- 258 00:17:08,940 --> 00:17:10,430 who knows? 259 00:17:10,430 --> 00:17:12,319 Two people, Adam and Eve. 260 00:17:12,319 --> 00:17:13,910 100 people, whatever. 261 00:17:13,910 --> 00:17:15,880 Whenever we start counting. 262 00:17:15,880 --> 00:17:18,500 This is time. 263 00:17:18,500 --> 00:17:20,609 You know, that's the time that we're starting. 264 00:17:20,609 --> 00:17:21,859 We could start today. 265 00:17:24,599 --> 00:17:30,640 This is time starting at some point where we know something. 266 00:17:30,640 --> 00:17:33,270 Well, we have pretty good numbers back for 267 00:17:33,270 --> 00:17:34,350 some hundreds of years. 268 00:17:34,350 --> 00:17:37,590 So probably it wouldn't be today we'd start. 269 00:17:37,590 --> 00:17:40,575 But wherever we start, we follow that equation. 270 00:17:44,560 --> 00:17:47,020 Let me just draw the graph. 271 00:17:47,020 --> 00:17:50,760 It grows. 272 00:17:50,760 --> 00:17:55,510 It grows and then at a certain point, and it turns out to be 273 00:17:55,510 --> 00:18:00,670 the halfway point, c over 2s. 274 00:18:00,670 --> 00:18:02,510 It's beautiful calculus. 275 00:18:02,510 --> 00:18:06,540 It turns out that that's a point of inflection. 276 00:18:06,540 --> 00:18:10,340 It's not only growing, it's growing faster and faster up 277 00:18:10,340 --> 00:18:11,420 to this time. 278 00:18:11,420 --> 00:18:15,740 I don't know when that is, but that's a very critical moment 279 00:18:15,740 --> 00:18:17,350 in the history of the world. 280 00:18:17,350 --> 00:18:20,400 I think we may be a little passed it actually. 281 00:18:20,400 --> 00:18:23,050 And after that point, the whole thing 282 00:18:23,050 --> 00:18:24,455 is just like symmetric. 283 00:18:24,455 --> 00:18:26,530 It's an s-shaped. 284 00:18:26,530 --> 00:18:34,090 It goes up and now slows down and gets closer and closer to 285 00:18:34,090 --> 00:18:36,960 this c/s steady state. 286 00:18:36,960 --> 00:18:39,010 I would call c/s a steady state. 287 00:18:39,010 --> 00:18:42,930 If it gets there it stays there steadily. 288 00:18:42,930 --> 00:18:45,010 It won't quite get there. 289 00:18:45,010 --> 00:18:46,340 This would be the s curve. 290 00:18:46,340 --> 00:18:49,440 And if I went back in time, it would be going back, back, 291 00:18:49,440 --> 00:18:52,000 back to nearly 0. 292 00:18:52,000 --> 00:18:53,580 Probably to 2. 293 00:18:53,580 --> 00:18:57,210 And up here, but not quite there. 294 00:18:57,210 --> 00:19:04,700 That's the population curve that comes from this model. 295 00:19:04,700 --> 00:19:07,510 I can't say that's the population curve that's going 296 00:19:07,510 --> 00:19:14,080 to happen in the next thousand years because the model is a 297 00:19:14,080 --> 00:19:14,840 good start. 298 00:19:14,840 --> 00:19:20,500 But of course, you could add in more terms for epidemics, 299 00:19:20,500 --> 00:19:24,760 for wars, for migration. 300 00:19:24,760 --> 00:19:31,140 All sorts of things affect the model. 301 00:19:31,140 --> 00:19:33,920 OK, now the math. 302 00:19:33,920 --> 00:19:36,940 The math says solve the equation. 303 00:19:36,940 --> 00:19:39,790 OK, how am I going to do that? 304 00:19:39,790 --> 00:19:42,580 Well, there are different ways, but here's 305 00:19:42,580 --> 00:19:44,250 a rather neat way. 306 00:19:44,250 --> 00:19:49,390 It turns out that if I try y. 307 00:19:49,390 --> 00:19:57,890 Let me introduce y as 1/P, then the equation for y is 308 00:19:57,890 --> 00:19:59,140 going to come out terrific. 309 00:20:02,240 --> 00:20:07,190 This is a trick that works for this equation. 310 00:20:07,190 --> 00:20:10,520 Nonlinear equations, it's OK to have a few tricks. 311 00:20:10,520 --> 00:20:15,500 All right, so this is y of t is 1/P of t. 312 00:20:15,500 --> 00:20:19,060 So of course the graph of y will be different and the 313 00:20:19,060 --> 00:20:21,030 equation for y will be different. 314 00:20:21,030 --> 00:20:24,590 Let me say what that equation looks like. 315 00:20:24,590 --> 00:20:27,810 OK, so I want to know, what's dy dt? 316 00:20:31,770 --> 00:20:34,010 I know how to take the derivative. 317 00:20:34,010 --> 00:20:35,590 We get to use calculus. 318 00:20:35,590 --> 00:20:38,220 This is P to the minus 1. 319 00:20:38,220 --> 00:20:43,500 So the derivative of P to the minus 1 is minus 1. 320 00:20:43,500 --> 00:20:47,905 P to the minus 2 times dP dt. 321 00:20:51,050 --> 00:20:52,680 That's the chain rule. 322 00:20:52,680 --> 00:20:56,160 P to the minus 1, the derivative is minus 1, p to 323 00:20:56,160 --> 00:21:00,060 the minus 2, and the derivative of P itself. 324 00:21:00,060 --> 00:21:02,800 And now I know dP dt. 325 00:21:02,800 --> 00:21:05,180 Now I'm going to use my equation. 326 00:21:05,180 --> 00:21:11,170 So that's cP minus sP squared with a minus. 327 00:21:11,170 --> 00:21:13,950 Oh, I can't lose that minus. 328 00:21:13,950 --> 00:21:15,920 Put it there. 329 00:21:15,920 --> 00:21:17,600 Keep your eye on that minus. 330 00:21:17,600 --> 00:21:19,185 Divided by P squared. 331 00:21:22,220 --> 00:21:24,470 So now I'll use the minus. 332 00:21:24,470 --> 00:21:28,170 That looks like an s to me. 333 00:21:28,170 --> 00:21:30,730 Minus minus is plus s. 334 00:21:30,730 --> 00:21:32,170 And what do I have here? 335 00:21:32,170 --> 00:21:34,340 Minus cP over P squared. 336 00:21:34,340 --> 00:21:42,580 I think I have minus c/P. And you say OK, true. 337 00:21:42,580 --> 00:21:44,790 But what good is that? 338 00:21:44,790 --> 00:21:48,890 But look, that 1/P is our y. 339 00:21:48,890 --> 00:21:55,470 So now I have s minus cy. 340 00:21:55,470 --> 00:22:01,550 The equation for y, the dy dt equation turns out to be 341 00:22:01,550 --> 00:22:07,710 linear with a source term s just as in the 342 00:22:07,710 --> 00:22:09,180 start of the lecture. 343 00:22:09,180 --> 00:22:18,790 And the growth rate term has a minus c, which we expect. 344 00:22:18,790 --> 00:22:24,450 Because our y is now 1 over. 345 00:22:24,450 --> 00:22:28,020 When this growth is going up exponentially, 1 over it is 346 00:22:28,020 --> 00:22:29,630 going down exponentially. 347 00:22:29,630 --> 00:22:33,280 And it turns out that same c. 348 00:22:33,280 --> 00:22:36,280 In other words, we can solve this equation. 349 00:22:36,280 --> 00:22:40,960 In fact, we have solved this equation just for-- 350 00:22:40,960 --> 00:22:44,370 shall I write down the answer? 351 00:22:44,370 --> 00:22:48,490 So the answer for y. 352 00:22:48,490 --> 00:22:54,240 You remember the answer for y is-- 353 00:22:54,240 --> 00:22:56,820 I'm going to look over here. 354 00:22:56,820 --> 00:22:59,400 Up there. 355 00:22:59,400 --> 00:23:03,130 This is our equation, the only difference is c is coming in 356 00:23:03,130 --> 00:23:03,970 with a minus sign. 357 00:23:03,970 --> 00:23:06,730 So I'm just going to write that same solution, 358 00:23:06,730 --> 00:23:09,110 just copy that here. 359 00:23:09,110 --> 00:23:11,280 y of t. 360 00:23:11,280 --> 00:23:15,920 Now c is coming in with a minus sign, is 361 00:23:15,920 --> 00:23:20,940 y of 0 minus s/c. 362 00:23:20,940 --> 00:23:22,950 c has that minus sign. 363 00:23:22,950 --> 00:23:26,050 e to the minus ct. 364 00:23:26,050 --> 00:23:27,300 OK, good. 365 00:23:30,560 --> 00:23:33,810 That's the solution for y. 366 00:23:37,230 --> 00:23:40,060 So we solved the equation. 367 00:23:40,060 --> 00:23:42,160 You could say, well, you solved the equation for y. 368 00:23:42,160 --> 00:23:45,890 But now P is just 1/y. 369 00:23:45,890 --> 00:23:54,130 Or y is 1/P. So I just go back now and change from y back to 370 00:23:54,130 --> 00:23:59,070 P. 1 over P of t is-- 371 00:23:59,070 --> 00:24:01,730 here I have 1 over P of 0. 372 00:24:06,840 --> 00:24:10,950 Is it OK if I leave the solution in that form? 373 00:24:10,950 --> 00:24:15,120 I can solve for P of t and get it all-- you get P of t equals 374 00:24:15,120 --> 00:24:17,140 and a whole lot of stuff on the right side. 375 00:24:17,140 --> 00:24:18,830 I move this over. 376 00:24:18,830 --> 00:24:21,850 I have to flip it upside down. 377 00:24:21,850 --> 00:24:25,040 It doesn't look as nice. 378 00:24:25,040 --> 00:24:26,390 Well, I guess I could. 379 00:24:26,390 --> 00:24:29,550 I could move that over and then just put 380 00:24:29,550 --> 00:24:32,100 to the minus 1 power. 381 00:24:32,100 --> 00:24:33,290 That would do it. 382 00:24:33,290 --> 00:24:36,270 And that is the solution. 383 00:24:36,270 --> 00:24:37,590 Maybe I should do that. 384 00:24:37,590 --> 00:24:40,250 P of t is-- 385 00:24:40,250 --> 00:24:41,410 I'm going to move this over. 386 00:24:41,410 --> 00:24:46,900 So this same parenthesis times e to the ct plus s/c. 387 00:24:46,900 --> 00:24:48,600 I moved that over. 388 00:24:48,600 --> 00:24:52,400 And then I say I have to flip it. 389 00:24:52,400 --> 00:24:55,210 So it's 1 over that. 390 00:24:55,210 --> 00:24:59,290 1 over all that stuff, where this is just this 391 00:24:59,290 --> 00:25:02,240 stuff copied again. 392 00:25:02,240 --> 00:25:05,450 So we get an expression but to me, that top one 393 00:25:05,450 --> 00:25:07,460 looks pretty nice. 394 00:25:07,460 --> 00:25:11,140 And graphing it is no problem. 395 00:25:11,140 --> 00:25:14,460 And that's what the graph looks like. 396 00:25:14,460 --> 00:25:18,520 It's that famous s curve. 397 00:25:18,520 --> 00:25:23,960 What we've solved here was the population equation and it's 398 00:25:23,960 --> 00:25:26,265 often called the logistic equation. 399 00:25:34,155 --> 00:25:37,790 I mention that word because you could see it there as the 400 00:25:37,790 --> 00:25:41,270 heading for this particular model. 401 00:25:41,270 --> 00:25:50,290 So let me let you leave open everything we've done today. 402 00:25:50,290 --> 00:25:56,890 But maybe I can say the most interesting aspect is the 403 00:25:56,890 --> 00:25:58,140 model itself. 404 00:26:00,290 --> 00:26:01,820 To what extent is it accurate? 405 00:26:01,820 --> 00:26:03,080 We could try it. 406 00:26:03,080 --> 00:26:08,480 We could estimate c and s, those numbers, to fit the 407 00:26:08,480 --> 00:26:12,980 model of what we know over one time period. 408 00:26:12,980 --> 00:26:15,140 And then we could see, does it extend over 409 00:26:15,140 --> 00:26:16,760 another time period? 410 00:26:16,760 --> 00:26:20,550 I was going to say about Wikipedia. 411 00:26:20,550 --> 00:26:23,360 And generally, Wikipedia is not too bad. 412 00:26:23,360 --> 00:26:26,775 They made one goof, which I thought was awful. 413 00:26:29,630 --> 00:26:34,950 The question was, what's the growth rate? 414 00:26:34,950 --> 00:26:37,350 Actually, Wikipedia doesn't discuss 415 00:26:37,350 --> 00:26:38,810 this particular equation. 416 00:26:38,810 --> 00:26:43,260 It tells you a lot of other things about population. 417 00:26:43,260 --> 00:26:45,600 Of course, they talk about the growth rate and at one point 418 00:26:45,600 --> 00:26:48,580 they say at one time, it was 1.8%. 419 00:26:51,890 --> 00:26:55,920 I say 1.8% is not a growth rate. 420 00:26:55,920 --> 00:26:59,040 Growth rate, c, don't have-- 421 00:26:59,040 --> 00:27:01,570 they're not percentages. 422 00:27:01,570 --> 00:27:04,380 Their units are 1 over time. 423 00:27:04,380 --> 00:27:08,370 And maybe I could make that point, emphasize that point. 424 00:27:08,370 --> 00:27:13,940 When I see ct together, that tells me that has the 425 00:27:13,940 --> 00:27:15,930 dimension of time, of course. 426 00:27:15,930 --> 00:27:20,010 So c must have the dimensions of 1 over time. 427 00:27:20,010 --> 00:27:25,410 The growth rate is 1.8% per year. 428 00:27:25,410 --> 00:27:29,360 And I will admit that three lines later on Wikipedia, when 429 00:27:29,360 --> 00:27:34,330 they're referring to an earlier growth rate earlier in 430 00:27:34,330 --> 00:27:42,700 the last century, they do say 2.2% per year. 431 00:27:42,700 --> 00:27:44,270 They get it right. 432 00:27:44,270 --> 00:27:48,570 The units are 1 over time. 433 00:27:48,570 --> 00:27:53,680 The solution is telling us that and the equation is 434 00:27:53,680 --> 00:27:54,380 telling us that. 435 00:27:54,380 --> 00:27:57,400 This has the dimensions of population, number of people. 436 00:27:57,400 --> 00:27:59,720 So does this. 437 00:27:59,720 --> 00:28:02,430 I'm dividing by a time. 438 00:28:02,430 --> 00:28:07,790 So c must be the source of that division by a time. 439 00:28:07,790 --> 00:28:10,050 The units of c must be 1 over time. 440 00:28:10,050 --> 00:28:15,850 You might like to figure out the units of s because in an 441 00:28:15,850 --> 00:28:18,840 equation, everything has to have the same units, the same 442 00:28:18,840 --> 00:28:23,260 dimension to make any sense at all. 443 00:28:23,260 --> 00:28:28,860 So with that little comment, let me emphasize 444 00:28:28,860 --> 00:28:32,740 the interest in just-- 445 00:28:32,740 --> 00:28:37,460 if you have a project to do, this could be a quite 446 00:28:37,460 --> 00:28:38,960 remarkable one. 447 00:28:38,960 --> 00:28:43,870 Now just I'll write down, but not solve-- 448 00:28:43,870 --> 00:28:46,720 since I have a little space and mathematicians tend to 449 00:28:46,720 --> 00:28:49,140 fill space-- 450 00:28:49,140 --> 00:28:53,150 I'll write down one more equation. 451 00:28:53,150 --> 00:28:55,930 Actually, it will be two equations because it's a 452 00:28:55,930 --> 00:29:00,150 predator and a prey. 453 00:29:00,150 --> 00:29:03,850 A hunter and a hunted. 454 00:29:03,850 --> 00:29:10,150 And in ecology, or modeling, whatever. 455 00:29:10,150 --> 00:29:13,080 Foxes and rabbits, maybe. 456 00:29:13,080 --> 00:29:17,220 We might have these equations, the predator 457 00:29:17,220 --> 00:29:21,520 and the prey equation. 458 00:29:21,520 --> 00:29:24,070 And what would that look like? 459 00:29:24,070 --> 00:29:25,510 So we have two unknowns. 460 00:29:25,510 --> 00:29:26,360 The predator. 461 00:29:26,360 --> 00:29:27,810 Can I call that u? 462 00:29:27,810 --> 00:29:29,060 So I have a du dt. 463 00:29:31,760 --> 00:29:35,080 What's the growth of the predators, the number or the 464 00:29:35,080 --> 00:29:35,890 population? 465 00:29:35,890 --> 00:29:40,300 These are both populations, but of two different species. 466 00:29:40,300 --> 00:29:45,170 And the prey will be v. So u is the 467 00:29:45,170 --> 00:29:46,515 population of the predator. 468 00:29:49,220 --> 00:29:53,270 If the predator has nothing to eat, the populations of 469 00:29:53,270 --> 00:29:55,840 predator is going to drop. 470 00:29:55,840 --> 00:29:59,650 But the more prey that's around-- 471 00:29:59,650 --> 00:30:03,150 so here we'll have a u times v going positive. 472 00:30:03,150 --> 00:30:12,280 So we'll have something like a minus cu as a term that if 473 00:30:12,280 --> 00:30:16,360 there's nothing to eat, if that's all there is, if the 474 00:30:16,360 --> 00:30:19,660 prey is all gone, the predator will die out. 475 00:30:19,660 --> 00:30:24,570 But when there is a predator, there will be a source term 476 00:30:24,570 --> 00:30:31,540 proportional to u and v. The more predators there are, 477 00:30:31,540 --> 00:30:33,460 they're all eating away. 478 00:30:33,460 --> 00:30:38,010 And the more prey there is the more they're eating. 479 00:30:38,010 --> 00:30:40,800 And now what about dv dt? 480 00:30:40,800 --> 00:30:43,910 OK, v is a totally different position. 481 00:30:43,910 --> 00:30:48,470 I guess it's getting eaten. 482 00:30:48,470 --> 00:30:53,930 So this term, well, I'm not sure what all the 483 00:30:53,930 --> 00:30:57,490 constants are here. 484 00:30:57,490 --> 00:31:00,550 The prey, this is now the prey, the little rabbits. 485 00:31:00,550 --> 00:31:02,470 They're just eating grass. 486 00:31:02,470 --> 00:31:04,620 Plenty of grass around. 487 00:31:04,620 --> 00:31:06,110 So they're reproducing. 488 00:31:06,110 --> 00:31:14,480 So they have some positive growth rate, capital C. 489 00:31:14,480 --> 00:31:16,540 Multiple per year. 490 00:31:16,540 --> 00:31:24,410 But then they will have a negative term as we saw in 491 00:31:24,410 --> 00:31:29,230 population coming from the competition with a predator. 492 00:31:29,230 --> 00:31:34,860 So this would be a model, a predator-prey model that also 493 00:31:34,860 --> 00:31:38,950 shows up in many applications of mathematics. 494 00:31:38,950 --> 00:31:44,510 So there you see two linear differential equations. 495 00:31:44,510 --> 00:31:48,310 The simple one dy dt we'll see y. 496 00:31:48,310 --> 00:31:51,080 We totally know how to solve that. 497 00:31:51,080 --> 00:31:56,460 Now with a source term s, we're OK, and we got the 498 00:31:56,460 --> 00:31:59,900 solution there and we found it a second way. 499 00:31:59,900 --> 00:32:06,040 Then thirdly was the main interest, the population 500 00:32:06,040 --> 00:32:09,520 equation, the logistic equation with a P squared, 501 00:32:09,520 --> 00:32:15,130 which we were able to solve and graph the solution. 502 00:32:15,130 --> 00:32:17,810 That s curve is just fantastic. 503 00:32:17,810 --> 00:32:22,980 If you want a challenge, check that this-- 504 00:32:22,980 --> 00:32:25,420 so what do I mean at that inflection point? 505 00:32:25,420 --> 00:32:26,960 What's happening there? 506 00:32:26,960 --> 00:32:35,260 I claim that at this value of P, the curve stops bending up, 507 00:32:35,260 --> 00:32:39,710 starts bending down, which means second 508 00:32:39,710 --> 00:32:45,710 derivative of P is 0. 509 00:32:45,710 --> 00:32:48,110 You could check that. 510 00:32:48,110 --> 00:32:52,280 Take the equation, take its derivative to get the second 511 00:32:52,280 --> 00:32:52,830 derivative. 512 00:32:52,830 --> 00:32:56,040 Then that'll involve dP dts over here. 513 00:32:56,040 --> 00:32:58,640 But then you know dP dt, so plug that in. 514 00:32:58,640 --> 00:33:01,810 And then finally, you'll get down to P's. 515 00:33:01,810 --> 00:33:05,670 And I think you'll find that the thing cancels itself out. 516 00:33:05,670 --> 00:33:10,200 The second derivative, the bending in the s curve, is 0 517 00:33:10,200 --> 00:33:11,790 at c over 2s. 518 00:33:11,790 --> 00:33:15,170 So that's the moment of fastest growth. 519 00:33:15,170 --> 00:33:22,780 And maybe that happened relatively recently. 520 00:33:22,780 --> 00:33:26,430 And we're slowing down, but we've got a long way to go. 521 00:33:26,430 --> 00:33:29,380 So roughly, I think the population now is 522 00:33:29,380 --> 00:33:32,630 just under 7 billion. 523 00:33:32,630 --> 00:33:36,270 And oh boy, if we just left-- 524 00:33:36,270 --> 00:33:39,850 if it was 7 billion at that point, it would be 14 billion 525 00:33:39,850 --> 00:33:41,480 at the end. 526 00:33:41,480 --> 00:33:44,350 I think we're a better further along now at 7 billion. 527 00:33:44,350 --> 00:33:45,840 7 billion's somewhere about here. 528 00:33:45,840 --> 00:33:51,880 So maybe the end with this model is more like 10 or 11 529 00:33:51,880 --> 00:33:58,700 billion as the population of the earth at t equal infinity. 530 00:33:58,700 --> 00:34:00,820 OK, that's a mathematical model. 531 00:34:00,820 --> 00:34:02,580 Thank you. 532 00:34:02,580 --> 00:34:04,390 ANNOUNCER: This has been a production of MIT 533 00:34:04,390 --> 00:34:06,780 OpenCourseWare and Gilbert Strang. 534 00:34:06,780 --> 00:34:09,060 Funding for this video was provided by the Lord 535 00:34:09,060 --> 00:34:10,270 Foundation. 536 00:34:10,270 --> 00:34:13,400 To help OCW continue to provide free and open access 537 00:34:13,400 --> 00:34:16,480 to MIT courses, please make a donation at 538 00:34:16,480 --> 00:34:18,040 ocw.mit.edu/donate.