1 00:00:11,240 --> 00:00:19,730 OK. Welcome to introductory calculus. I will start with some practical information, 2 00:00:19,730 --> 00:00:24,860 and then I'll tell you a little bit about the syllabus and what we will cover in this 3 00:00:24,860 --> 00:00:35,560 course and then give you some examples of differential equations from physical sciences. 4 00:00:35,560 --> 00:00:46,570 And then a little bit of integration towards the end, so for for many of you, this might be the easiest course here that you take in Oxford. 5 00:00:46,570 --> 00:00:49,930 But I think things will get progressively harder. 6 00:00:49,930 --> 00:00:58,330 So maybe in a couple of weeks, it will be interesting to everybody if today's lecture might be a bit too easy for some of you. 7 00:00:58,330 --> 00:01:15,090 OK, so let me tell you some practical information. So we have 16 lectures. 8 00:01:15,090 --> 00:01:35,030 The lecture notes are online. Online, these are the lecture notes, these were written by Kath Wilkins. 9 00:01:35,030 --> 00:01:46,730 She taught this course for a few years until last year, so we'll just follow them. 10 00:01:46,730 --> 00:01:57,390 I guess I should have introduced myself at some points of the lecture in. 11 00:01:57,390 --> 00:02:06,150 So you can call me down, my name is Daniel thorugh and will meet on. 12 00:02:06,150 --> 00:02:19,450 So we'll meet twice a week. Today's special just because he's the first week we'll meet on Mondays and Wednesdays at 10 a.m., So not too early. 13 00:02:19,450 --> 00:02:30,560 And you'll have eight problem sheets. So the first two problem sheets are online. 14 00:02:30,560 --> 00:02:41,980 The problem sheets you'll cover in for tutorials in your college. 15 00:02:41,980 --> 00:02:48,090 OK, so for our our tutorial. 16 00:02:48,090 --> 00:02:57,570 What do I? So I said the lecture notes on a line, the reading list is also line. 17 00:02:57,570 --> 00:03:12,420 So see online. The book that I like is Mary Boluses Mathematical Methods in physical sciences. 18 00:03:12,420 --> 00:03:33,920 You know this book? And most of your colleagues should have a copy of of this, if not, the university does as well. 19 00:03:33,920 --> 00:03:42,950 So this this book is quite concise, and it has various examples from physics and engineering and science, 20 00:03:42,950 --> 00:03:48,980 and it also has the added advantage that if if unlike the other books on on the reading list, 21 00:03:48,980 --> 00:03:59,600 if you drop this one on your foot, you might be able to to walk without seeing an orthopaedic surgeon. 22 00:03:59,600 --> 00:04:08,390 All right, so that's any any questions about this? 23 00:04:08,390 --> 00:04:30,010 OK, now, syllabus. So the first half of the course, about about seven or eight lectures will be devoted to differential equations. 24 00:04:30,010 --> 00:04:57,280 So this is about seven eight lectures. So if two kinds ordinary differential equations, all these and partial differential equations. 25 00:04:57,280 --> 00:05:12,640 Our is. So I'll give you I'll give you some examples very soon, we'll look at fairly easy examples of differential equations. 26 00:05:12,640 --> 00:05:17,140 We'll learn some techniques. It's it's a combination solving them. 27 00:05:17,140 --> 00:05:30,040 It's a combination of science and art. You have to do some educated guesses at some point, but it's it's quite an interesting and very useful subject. 28 00:05:30,040 --> 00:05:41,560 And then after that, we'll talk about lying and double integrals, line integrals and double intervals. 29 00:05:41,560 --> 00:05:50,910 And the reason these are useful is because we will be able to compute arc length. 30 00:05:50,910 --> 00:05:57,480 Of of cars and areas. 31 00:05:57,480 --> 00:06:04,160 Various regions in the plane or surfaces. 32 00:06:04,160 --> 00:06:22,550 So this is maybe three lectures. And then finally, we'll do calculus of functions. 33 00:06:22,550 --> 00:06:34,070 In two variables. So this should be viewed as a gentle introduction into multivariable calculus. 34 00:06:34,070 --> 00:06:45,980 So amongst amongst the things that will do, we'll look at various surfaces, gradients, normal vectors. 35 00:06:45,980 --> 00:07:03,580 We'll look at Taylor's theorem into variables, critical points and a little bit of Lagrange multipliers. 36 00:07:03,580 --> 00:07:09,790 Which are useful for optimisation problems. OK. 37 00:07:09,790 --> 00:07:17,890 Now, there is a lot of interaction between this course and other brilliant courses that you will take. 38 00:07:17,890 --> 00:07:26,900 So intro calculus. Well, be directly useful. 39 00:07:26,900 --> 00:07:35,530 Well, obviously multivariable calculus, as I said. 40 00:07:35,530 --> 00:07:43,780 In a way, it's a little bit unfair. We we set set up the work we do some examples for in introduction calculus. 41 00:07:43,780 --> 00:07:48,670 But then the really cool results and theorems you prove in multivariable calculus. 42 00:07:48,670 --> 00:07:54,040 So we just do a little bit of the groundwork towards that. 43 00:07:54,040 --> 00:08:09,850 You also do. These are also useful in dynamics and in B, the ease of you will do the next term for the series and please. 44 00:08:09,850 --> 00:08:19,960 Now there is a lot of interaction between internal calculus and analysis, particularly analysis tool, which is what you do next term. 45 00:08:19,960 --> 00:08:30,080 So there will be quite a few results. From analysis that will just fade and not prove, maybe prove some particular examples and so on. 46 00:08:30,080 --> 00:08:36,590 But real rigorous proof seal doing analysis next term. 47 00:08:36,590 --> 00:08:46,590 But then it all comes together when you revise or for your exams in Trinity. 48 00:08:46,590 --> 00:08:51,170 OK. So that's that, of course, in part A. 49 00:08:51,170 --> 00:09:00,110 There will be lots of applied mathematics options that will continue this differential equations is a big option, fluid and waves, et cetera. 50 00:09:00,110 --> 00:09:05,210 So this is a very useful course. It's also mandatory. So you have you have to be. 51 00:09:05,210 --> 00:09:24,550 Yes. OK. So now let me give you some examples of where all these might appear. 52 00:09:24,550 --> 00:09:36,300 OK, so all these. So what what is a different ordinary differential equation? 53 00:09:36,300 --> 00:10:03,300 So this is an equation. Involving an independent variable, let's call it X and a function of X. 54 00:10:03,300 --> 00:10:28,310 Which we call it. Why? So why this would be the dependent variable and the derivatives of why? 55 00:10:28,310 --> 00:10:40,100 With respect to X. So for example, DUI, the X D squared, y squared, 56 00:10:40,100 --> 00:10:51,440 etc. So the order of the highest derivative that occurs when you call that the order of the differential equation. 57 00:10:51,440 --> 00:11:07,680 So, for example, the simplest so the simplest kind of, Audie, would be something of the foreign DUI. 58 00:11:07,680 --> 00:11:16,020 The X equals some function in X. 59 00:11:16,020 --> 00:11:37,040 So the wide X equals four works, you can solve that by direct integration, so this can be solved. 60 00:11:37,040 --> 00:11:46,460 So y equals so y you can think of Y as being the entity derivative of F of X and then we can use integration. 61 00:11:46,460 --> 00:11:53,480 That's the simplest kind of differential equation that we can have, 62 00:11:53,480 --> 00:12:02,090 and this is the reason why we'll start the course by reviewing a little bit of integration techniques. 63 00:12:02,090 --> 00:12:06,890 But more interesting, there could be more interesting differential equations. 64 00:12:06,890 --> 00:12:15,550 So let me give you some examples from. From physical sciences. 65 00:12:15,550 --> 00:12:24,430 So, for example, from mechanics, this is something that you have all seen. 66 00:12:24,430 --> 00:12:32,880 You can have Newton's second law. 67 00:12:32,880 --> 00:12:46,810 Which says that the force is the mass times, the acceleration, so a is the acceleration. 68 00:12:46,810 --> 00:13:03,420 But then the acceleration is a derivative, is the derivative of the velocity with respect to time. 69 00:13:03,420 --> 00:13:09,240 So that's already a differential equation, but it could be a second or that differential equation, 70 00:13:09,240 --> 00:13:24,230 if you think that V is B R B T, where R is the displacement? 71 00:13:24,230 --> 00:13:36,520 Then you get, for example, a is the R v squared r d squared, which is a second order differential equation so that. 72 00:13:36,520 --> 00:13:42,840 That's an easy example of how differential equations appear in mechanics. 73 00:13:42,840 --> 00:13:49,370 Well, you could also have differential equations in. 74 00:13:49,370 --> 00:14:01,180 Of engineering or if you have an electrical circuit. 75 00:14:01,180 --> 00:14:14,380 So if I take a simple one, so a simple series circuit, which, for example, an R L C circuit, 76 00:14:14,380 --> 00:14:26,480 which means that it has the following components it has R stands for resistor, so it has a registered R. 77 00:14:26,480 --> 00:14:42,670 It has an inductor L with Inductance L, and it has a capacitor with the party capacitance C. 78 00:14:42,670 --> 00:15:02,460 And it has a source of. Voltage, something like a battery v, so here I am, so I have a capacitor with C capacitance. 79 00:15:02,460 --> 00:15:20,660 And the register. With our assistance and an inductor with L and inductance. 80 00:15:20,660 --> 00:15:33,060 So here are our L and C our constants, they're independent of time. 81 00:15:33,060 --> 00:15:42,380 But then I'm interested, for example, in the current across the circuit, so this is the current. 82 00:15:42,380 --> 00:15:57,770 Across I of T is the current across the circuit, which is a function of the time. 83 00:15:57,770 --> 00:16:02,930 So in terms of differential equations, t the time would be the independent variable. 84 00:16:02,930 --> 00:16:07,550 And this i of T, for example, is a dependent variable. 85 00:16:07,550 --> 00:16:16,660 I can also have Q of T, which is the charge across. 86 00:16:16,660 --> 00:16:23,040 Capacitor. Sorry. 87 00:16:23,040 --> 00:16:36,140 On the capacitor and the relation between the two of them is that I is the Q, the T. 88 00:16:36,140 --> 00:16:45,340 So kick-offs law says that. 89 00:16:45,340 --> 00:17:00,450 The total voltage is zero around the circuit, which in another way, the voltage V from the battery, which is a function of T. 90 00:17:00,450 --> 00:17:10,410 Equals the voltage across the resistor, plus the voltage across the inductive plus the voltage on the capacitor, 91 00:17:10,410 --> 00:17:31,280 and now you're right, each one of them. The voltage across the resistor by ohms law is R times-I of T. 92 00:17:31,280 --> 00:17:40,210 The one on the capacitor is just one over three times the charge. 93 00:17:40,210 --> 00:17:52,980 And for the inductor is El the constant times DADT, which is faraday's. 94 00:17:52,980 --> 00:18:05,780 Hello. So now I can express, for example, so I have an equation involving V I. 95 00:18:05,780 --> 00:18:13,730 And Q, but I is the Q DP so I can rewrite everything in terms of Q, for example. 96 00:18:13,730 --> 00:18:28,860 So I can get a differential equation in Q, which will be simply. 97 00:18:28,860 --> 00:18:43,440 So this would be the leading term DDT, so l times the IDP becomes L times d squared, humidity squared, plus our eye is our times. 98 00:18:43,440 --> 00:18:57,940 Cue the 80 plus one over C times Q equals V, so that is a second order differential equation. 99 00:18:57,940 --> 00:19:05,440 That appears in electrical circuits, you. 100 00:19:05,440 --> 00:19:10,690 So it's second order, because the highest derivative is of second order. 101 00:19:10,690 --> 00:19:18,010 It has constant coefficients because the constants are L, R and C are constant. 102 00:19:18,010 --> 00:19:25,570 And it's what we'll call in homogeneous because this doesn't have to be zero. 103 00:19:25,570 --> 00:19:32,920 So those are the type of differential equations that we can we can study. 104 00:19:32,920 --> 00:19:41,740 And there are many other examples, so I'll leave one as an exercise for you. 105 00:19:41,740 --> 00:19:59,790 And is the exercise? So I'll tell you, the problem is the rate at which a radioactive. 106 00:19:59,790 --> 00:20:06,640 Substance, because. 107 00:20:06,640 --> 00:20:19,060 Is proportional. To the remaining. 108 00:20:19,060 --> 00:20:26,450 A number of. Number of attempts. 109 00:20:26,450 --> 00:20:35,380 So I want you to as an exercise to write the differential equation that describes this situation. 110 00:20:35,380 --> 00:20:40,630 OK, so we'll come back to things like this later. 111 00:20:40,630 --> 00:20:58,550 So what? So the question is, what's the what's the differential equation? 112 00:20:58,550 --> 00:21:08,600 OK. So as as you progress along in this course, in the mathematics course here you will encounter very, 113 00:21:08,600 --> 00:21:15,350 very interesting and sophisticated differential equations in applied mathematics. 114 00:21:15,350 --> 00:21:25,970 So we're just scratching the surface a little. All right, now, going back to what I what I said before, 115 00:21:25,970 --> 00:21:34,970 the simplest kind of ODA is the idea X equals f of X, which you can solve by direct integration. 116 00:21:34,970 --> 00:21:41,680 So let me review a couple of facts about integration. 117 00:21:41,680 --> 00:22:03,470 So one of the most useful techniques, which I'm sure most of you are quite familiar with is integration by parts. 118 00:22:03,470 --> 00:22:14,120 OK, so where does integration by parts come from? Well, it comes from the product rules, product or life needs rule. 119 00:22:14,120 --> 00:22:26,850 If you want to sound fancy. For derivatives. 120 00:22:26,850 --> 00:22:35,670 So if I have two functions, FMG and I multiply them and then I differentiate them, so I have four times. 121 00:22:35,670 --> 00:22:41,570 Prime is f prime g plus f g prime. 122 00:22:41,570 --> 00:22:56,290 Which means that. F g f times, g prime equals f times g prime minus f prime times G. 123 00:22:56,290 --> 00:23:09,200 And if I integrate both sides. Then I end up with the integration by parts, which is. 124 00:23:09,200 --> 00:23:18,040 F times, G Prime B X, if they're functions of X, equals F. 125 00:23:18,040 --> 00:23:25,260 F times, g minus f prime times g, d x. 126 00:23:25,260 --> 00:23:30,660 OK, so this is the version, the indefinite integrals version. 127 00:23:30,660 --> 00:23:40,400 You can have a definite intervallo version where you put the limits of integration. 128 00:23:40,400 --> 00:23:54,720 So let me spell it out. So this is the definite into integrals version. 129 00:23:54,720 --> 00:24:08,800 All right. So let's do a couple of examples. 130 00:24:08,800 --> 00:24:23,800 So the first example. So suppose I want to integrate X squared sign X. 131 00:24:23,800 --> 00:24:37,330 The. Now. So this would solve so this would give this gives the solution. 132 00:24:37,330 --> 00:24:47,020 To DUI, the X equals X squared sine X. 133 00:24:47,020 --> 00:24:52,900 OK, so integration by parts, you have to decide which one is off and which ones? 134 00:24:52,900 --> 00:25:02,440 G. Now clearly I would like to decrease. The power here, I know I can never get rid of the sign by differentiation. 135 00:25:02,440 --> 00:25:15,590 So then maybe this, then I have to do this Earth and this is G Prime, which means that G is minus cos x. 136 00:25:15,590 --> 00:25:22,590 So if I call this interval, I I is. 137 00:25:22,590 --> 00:25:33,470 X squared times minus six and then minus the derivative of F, which is two x times minus cos x. 138 00:25:33,470 --> 00:25:49,160 The ex. So this is minus x squared cos x plus two times x x x. 139 00:25:49,160 --> 00:25:57,140 And now again, this should be F and this should be G Prime. 140 00:25:57,140 --> 00:26:02,230 Score across X plus two times. 141 00:26:02,230 --> 00:26:09,720 X Sign X. Minus two times. 142 00:26:09,720 --> 00:26:18,190 Sign X the X, so please try to follow through what I'm doing and let me know if I make a mistake, 143 00:26:18,190 --> 00:26:25,810 this is this is kind of my nightmare to do integrate into clubs like this while I'm being filmed. 144 00:26:25,810 --> 00:26:40,620 This is not exactly what I like to do. So two x sign x and then minus korsak, then plus C. 145 00:26:40,620 --> 00:26:48,350 Is this. So, so. Plus plus. 146 00:26:48,350 --> 00:26:54,460 Thank you. Good. As I said. 147 00:26:54,460 --> 00:26:59,750 So see here, denotes. A constant. 148 00:26:59,750 --> 00:27:26,160 Because we're doing indefinite integral. All right, let's do another example. 149 00:27:26,160 --> 00:27:33,780 So again, an indefinite integral to X minus one times Alan X squared plus one. 150 00:27:33,780 --> 00:27:42,300 The X. OK. What do you think? 151 00:27:42,300 --> 00:27:54,050 How which one should be f and which ones should be G or G Prime? 152 00:27:54,050 --> 00:27:58,730 Say that again. Right. 153 00:27:58,730 --> 00:28:05,810 So this I want to differentiate to get rid of the logger, so I should call this, which means that this is going to be different. 154 00:28:05,810 --> 00:28:13,560 Thank you. And that makes G X squared minus six. 155 00:28:13,560 --> 00:28:22,530 So this becomes X squared minus six times l n x squared plus one minus the integral. 156 00:28:22,530 --> 00:28:38,010 Of X squared minus six times the derivative of the natural log of X squared plus one, which is two x over x squared plus one d x. 157 00:28:38,010 --> 00:28:48,180 So I'm finally this term. What do I do here? 158 00:28:48,180 --> 00:28:55,170 Good. So we do long division. So let's rewrite it first. 159 00:28:55,170 --> 00:29:08,430 This is x squared minus x l n x squared plus one and then minus two x cubed minus x squared over x squared plus one d x. 160 00:29:08,430 --> 00:29:13,650 So I have to remember how to do long division. 161 00:29:13,650 --> 00:29:20,230 So how x cubed minus x. Now. 162 00:29:20,230 --> 00:29:26,290 Depending how you learn this, you will draw the long division in different ways. 163 00:29:26,290 --> 00:29:31,360 So you just do it your way, and I'll I'll do it my way. 164 00:29:31,360 --> 00:29:44,030 So that's X and minus six cubed. Minus X then. 165 00:29:44,030 --> 00:29:55,860 Well, minus x squared minus six, and that's minus one. 166 00:29:55,860 --> 00:30:08,250 OK, so this means that x cubed minus x squared over x squared plus one equals x minus one. 167 00:30:08,250 --> 00:30:16,350 Plus, minus six plus one over x squared plus one. 168 00:30:16,350 --> 00:30:24,090 Did you get the same thing? Good. OK, so let's call this integral, Jay. 169 00:30:24,090 --> 00:30:38,840 And now we come to you, Jay. The integral of X minus one plus or minus X plus one over x squared plus one. 170 00:30:38,840 --> 00:30:48,860 The X, which equals X squared minus half x squared minus X. 171 00:30:48,860 --> 00:30:54,760 And then. How do I integrate this term? 172 00:30:54,760 --> 00:31:07,730 The fraction? So I should split x over x squared plus one. 173 00:31:07,730 --> 00:31:12,950 The ex. Yeah. 174 00:31:12,950 --> 00:31:21,910 And let me write the last term plus. The X over x squared plus one. 175 00:31:21,910 --> 00:31:27,440 So this one, the last term, we should recognise that what is it? 176 00:31:27,440 --> 00:31:31,620 Are 10 or 10 years, depending how you want to. 177 00:31:31,620 --> 00:31:42,860 The notices are 10 of X, just 10 inverse of X and what do we do with this? 178 00:31:42,860 --> 00:31:54,650 We can substitute, yeah, let let, let's do that so that we remember how to do substitutions. 179 00:31:54,650 --> 00:32:03,680 You might just look at it and know what it is, right, but just to review substitution if I said you equals x squared plus one. 180 00:32:03,680 --> 00:32:15,500 Then deal equals to X the X deal, the X equals to X, which means that this is one half. 181 00:32:15,500 --> 00:32:24,220 But the you over you. Which is one half line of you. 182 00:32:24,220 --> 00:32:31,700 She is one half a line of x squared plus one that you might have guessed. 183 00:32:31,700 --> 00:32:37,600 Just because you have enough practise, some of you. 184 00:32:37,600 --> 00:32:45,740 OK, so now let's put them all together, so Jay is one half x squared minus X. 185 00:32:45,740 --> 00:32:51,980 Minus one half, Alan X squared plus one. 186 00:32:51,980 --> 00:32:58,840 Plus. Ten inverse of X and some constant. 187 00:32:58,840 --> 00:33:04,440 Which means that the original integral, the. 188 00:33:04,440 --> 00:33:20,380 The integral in the beginning. Which I should have called I so that I don't have to roll down the balls but equals. 189 00:33:20,380 --> 00:33:26,740 X squared minus x l and x squared. 190 00:33:26,740 --> 00:33:39,320 Plus one. Minus twice this, so minus x squared plus two x plus Ellen. 191 00:33:39,320 --> 00:33:46,540 X squared plus one. Minus 10 in verse six and then. 192 00:33:46,540 --> 00:33:51,660 Plus to see. Thank you. 193 00:33:51,660 --> 00:34:06,530 Any other mistakes? All right. 194 00:34:06,530 --> 00:34:13,880 OK. So that's a that's an integral. 195 00:34:13,880 --> 00:34:21,710 There are cases when integration by parts will not simplify either of the two functions FMG. 196 00:34:21,710 --> 00:34:26,350 But what happens is if you do it twice, then you sort of come back to. 197 00:34:26,350 --> 00:34:33,610 What you started with, so the typical example. And. 198 00:34:33,610 --> 00:34:48,890 Is. I equals the to e to the X, Senex, the X. 199 00:34:48,890 --> 00:34:56,430 All right. So maybe we don't need to go through the entire calculation. 200 00:34:56,430 --> 00:35:08,210 This is in the lecture notes as well. But how would you solve it? 201 00:35:08,210 --> 00:35:18,070 Right. So you do you do it? So for example, I can say that this is gee prime and this is off. 202 00:35:18,070 --> 00:35:25,240 And then I integrate, I get calls and then I do it again, and I will end up with some expression minus. 203 00:35:25,240 --> 00:35:27,790 They seem to grow and then I sold for it. 204 00:35:27,790 --> 00:35:39,490 So you, you do this and you get the answer to be something like one half e to the X sign X minus six, then plus. 205 00:35:39,490 --> 00:35:52,090 Constant. OK. 206 00:35:52,090 --> 00:35:59,530 So another type of examples, which is which are more difficult, 207 00:35:59,530 --> 00:36:06,080 are the ones which you cannot solve in just one go, but you have to find a recursive formula. 208 00:36:06,080 --> 00:36:13,240 So I'll just do an example like that. You've you've seen other examples before. 209 00:36:13,240 --> 00:36:31,470 So this is when we get a reduction or if you want to call it a recursive formula. 210 00:36:31,470 --> 00:36:38,820 So I start I suppose, I'm looking at this interval. 211 00:36:38,820 --> 00:36:43,600 Co-sign to the end, ex, the ex. 212 00:36:43,600 --> 00:36:59,850 Now I want to label this integral I n because I'm going to get a formula of I n in terms of i n minus one or I n minus two, et cetera. 213 00:36:59,850 --> 00:37:04,160 Now, there is not much choice here, what you should call and. 214 00:37:04,160 --> 00:37:15,800 Mother, you should call. So I'm going to just do it, so this is coarse and minus one x times call six the X. 215 00:37:15,800 --> 00:37:27,280 So this is Earth and this is g prime. Then we get costs and minus one x sine x minus. 216 00:37:27,280 --> 00:37:41,780 The integral now I need to differentiate, if so and minus one costs and minus two x and then minus sine X. 217 00:37:41,780 --> 00:37:55,820 And then another sign X the X. It equals costs and minus one x sine X. 218 00:37:55,820 --> 00:38:01,940 Minus and minus one times, or maybe I'll make it a plus. 219 00:38:01,940 --> 00:38:12,350 Costs. And minus two x sine squared x the X. 220 00:38:12,350 --> 00:38:19,780 So if I write it like that. What do you do now? 221 00:38:19,780 --> 00:38:28,000 He right sine squared as one minus cos squared X. 222 00:38:28,000 --> 00:38:35,100 Which then gives you costs and minus one x sine X. 223 00:38:35,100 --> 00:38:42,670 Plus and minus one, the integral, of course, and minus two x the X. 224 00:38:42,670 --> 00:38:52,920 Minus and minus swan, the interval, of course, and Square X the X. 225 00:38:52,920 --> 00:39:02,720 So now I recognise that this is the integral, of course, and minus two is I sob and minus two. 226 00:39:02,720 --> 00:39:12,250 And the integral, of course, and this is in. So I have an equal that. 227 00:39:12,250 --> 00:39:29,510 So if I sold for, I end. We got I and equals. 228 00:39:29,510 --> 00:39:42,610 So I get an I and equals course and minus one x sine X plus and minus one i n minus two. 229 00:39:42,610 --> 00:40:07,310 Which gives me the recursive formula. I n equals one over n course and minus one x sine x plus and minus one over n i minus. 230 00:40:07,310 --> 00:40:28,270 So this is true for all and greater than or equal to two. 231 00:40:28,270 --> 00:40:37,560 OK. Now, if I want to know all of these integrals, I mean, using this formula. 232 00:40:37,560 --> 00:40:48,610 What else do I need to know? I zero and I one because it drops down by two. 233 00:40:48,610 --> 00:41:10,450 So let's compute zero and one. So we also need I zero, and I won, so I zero would be just the integral the X, which is X plus a c. 234 00:41:10,450 --> 00:41:22,150 And I Wan is the integral, of course, x the X, which is signing X plus C. 235 00:41:22,150 --> 00:41:28,030 And now with this, you can you can get any integral you want. 236 00:41:28,030 --> 00:41:33,700 For example, if you want to get, I don't know, ice. 237 00:41:33,700 --> 00:41:49,490 I six. You just follow that and you get that it's one sixth course. 238 00:41:49,490 --> 00:42:00,720 To the fifth. Sine X plus five over six times I for. 239 00:42:00,720 --> 00:42:08,310 Which is one of the six course. Sine X plus five over six times. 240 00:42:08,310 --> 00:42:14,630 I four is one fourth. Of course. 241 00:42:14,630 --> 00:42:20,130 Cubed X Sine X Plus. 242 00:42:20,130 --> 00:42:39,840 Three, four, two. Then what's eye to eye to his one half cos x sign x plus one half I zero i zero six. 243 00:42:39,840 --> 00:42:53,390 So you put. You substitute this in there and they get an I six is one sixth course to the fifth sine x plus. 244 00:42:53,390 --> 00:42:59,040 Five over 24 cores, cubed sine X. 245 00:42:59,040 --> 00:43:15,990 Plus. Five times, three times, one over six times, four times to cause sine X plus. 246 00:43:15,990 --> 00:43:31,190 Five times three over. I want over six times, four times to X Plus see. 247 00:43:31,190 --> 00:43:41,870 The. So it has I think I think you can you can probably cook up a general formula using this example, you see how it goes. 248 00:43:41,870 --> 00:43:55,390 So if I asked you to write. A Psalm involving all the terms, I think you can you can get the coefficients of each time inductively. 249 00:43:55,390 --> 00:44:05,270 Good. OK, so this is a quick review of integration by parts if if you're not. 250 00:44:05,270 --> 00:44:11,530 Fully comfortable with these examples or similar examples, then get up, 251 00:44:11,530 --> 00:44:19,130 get together an integration textbook and do a few more examples with integration by parts, substitutions and so on. 252 00:44:19,130 --> 00:44:25,040 Because differential equity, what we will do in solving differential equations will learn a lot of techniques. 253 00:44:25,040 --> 00:44:31,610 But ultimately you will have to integrate some, some some functions, so you should be able to do that. 254 00:44:31,610 --> 00:44:38,120 So what we learnt is how do you how to reduce the problem to integrating various functions? 255 00:44:38,120 --> 00:44:54,300 But you'll have to be able to do that. OK, so we discussed about the simplest kind of odds. 256 00:44:54,300 --> 00:45:07,460 Which can be solved just by direct integration. The next simplest. 257 00:45:07,460 --> 00:45:22,430 All these. Are the so-called separable. 258 00:45:22,430 --> 00:45:44,390 Oh, these. So we had the case did X equals f of X, which you can just integrate, the next case would be the I.D. X equals A of X times B of Y. 259 00:45:44,390 --> 00:45:56,560 So what I mean by that is that this is a function only index. 260 00:45:56,560 --> 00:46:02,840 Only, and similarly, B of Y is a function of. 261 00:46:02,840 --> 00:46:18,620 Why only? If you have a situation like that, then you can reduce it to the direct integration with one simple trick. 262 00:46:18,620 --> 00:46:39,060 If B y is not zero, then you divide by it and you get one over B of y d y the X equals a x. 263 00:46:39,060 --> 00:46:50,420 And now you can integrate just as we did before. 264 00:46:50,420 --> 00:47:02,840 So you'll get then the integral. Yes. 265 00:47:02,840 --> 00:47:09,980 So the left hand side is the integral DUI over B of Y and the right hand side is a D X two. 266 00:47:09,980 --> 00:47:15,550 And now you have to direct integration switch. 267 00:47:15,550 --> 00:47:22,900 Hopefully we can we can solve the type of integrals that we have in this course will be the 268 00:47:22,900 --> 00:47:28,120 time for which you can apply integration by parts or some other techniques and solve them. 269 00:47:28,120 --> 00:47:40,840 If I if I were to write an arbitrary function there and ask you how to integrate it, then we can't do that in in a closed formula. 270 00:47:40,840 --> 00:48:02,010 OK, so here here's an example. Find the general solution. 271 00:48:02,010 --> 00:48:11,590 To the separable. Differential equation. 272 00:48:11,590 --> 00:48:17,590 So the hint is already in the problem that this is a separable differential equation 273 00:48:17,590 --> 00:48:29,450 x times y squared minus one plus y x squared minus one d y d x equals two zero. 274 00:48:29,450 --> 00:48:35,340 And. X is between zero and one to avoid. 275 00:48:35,340 --> 00:48:46,690 Some some issues. About continuity or what not? 276 00:48:46,690 --> 00:48:54,290 OK. How do you separate this differential equation? 277 00:48:54,290 --> 00:49:04,550 I'll do separate variables. Correct. 278 00:49:04,550 --> 00:49:10,890 So I think you're. About two steps ahead of me, so I will first. 279 00:49:10,890 --> 00:49:14,190 But it's correct. But let me do it step by step. 280 00:49:14,190 --> 00:49:18,500 So what I will do is first isolate that. 281 00:49:18,500 --> 00:49:25,320 So I have y x squared minus one d y x equals. 282 00:49:25,320 --> 00:49:29,670 Minus x y squared minus one. 283 00:49:29,670 --> 00:49:39,370 And then separate the variables, as the name suggests, you have y over. 284 00:49:39,370 --> 00:50:12,150 Y squared minus one DUI, d x equals minus x over x squared minus Y. 285 00:50:12,150 --> 00:50:24,900 OK. What do we do now? Correct. 286 00:50:24,900 --> 00:50:30,020 So if we look at this, then so we integrate, let's integrate. 287 00:50:30,020 --> 00:50:48,540 Well, let me write one one more, so we integrate this and we get why over y squared minus one d y equals minus x over x squared minus one d x. 288 00:50:48,540 --> 00:50:55,080 So now we could do a substitution, as we did before, but I think we know how to do it. 289 00:50:55,080 --> 00:51:03,320 This looks like the derivative of a logarithm. So if I differentiate a land of X squared minus one. 290 00:51:03,320 --> 00:51:10,100 Then I get to X over X squared minus one. 291 00:51:10,100 --> 00:51:24,190 So except X is between zero and one, so maybe it's better to write this as one x over one minus x squared and get get rid of the minus. 292 00:51:24,190 --> 00:51:34,990 Sign. So then I'll do one minus x squared minus two x over one minus x squared. 293 00:51:34,990 --> 00:51:45,540 So then this is minus 11 of one minus x squared. 294 00:51:45,540 --> 00:51:53,320 And the half. And then plus, see? 295 00:51:53,320 --> 00:51:59,090 Whereas here I'll have to put absolute values because I don't know, it's one half. 296 00:51:59,090 --> 00:52:04,760 It's a long way squared. Minus one. 297 00:52:04,760 --> 00:52:17,290 In absolute value. Right. Now, the easiest way to write this is to get rid of the logarithm by moving this to the other side, 298 00:52:17,290 --> 00:52:22,040 using the properties of the logarithm an exponential rate. 299 00:52:22,040 --> 00:52:40,150 So let's do that. So we have one half. If I move the logarithm in X to the left hand side, then I use the property. 300 00:52:40,150 --> 00:52:50,440 Well, it doesn't matter much. Equals C. 301 00:52:50,440 --> 00:53:00,730 Which means that. The equation will be y squared minus one times one minus six squared. 302 00:53:00,730 --> 00:53:08,160 Absolute value equals it would be easy to see squared. 303 00:53:08,160 --> 00:53:15,100 Or E to to see which I can just call. 304 00:53:15,100 --> 00:53:25,950 Another kind of sea. And this would be a positive no. 305 00:53:25,950 --> 00:53:46,630 So the equation then that we get is. So the answer then is. 306 00:53:46,630 --> 00:53:52,880 Is that? Where a sea. 307 00:53:52,880 --> 00:53:57,500 It's positive, but I can't relax that this is always positive, 308 00:53:57,500 --> 00:54:04,430 because one minus x squared is always positive because I'm assuming X is between zero and one. 309 00:54:04,430 --> 00:54:10,300 But I can rewrite the answer in a nicer form by dropping. 310 00:54:10,300 --> 00:54:17,360 The absolute value and requiring and dropping the assumption on C. 311 00:54:17,360 --> 00:54:26,980 So another way of doing this is. 312 00:54:26,980 --> 00:54:41,740 Well, for you uniformity, I'll write it as one minus y squared equals C, so one minus y squared times one minus x squared equals C. 313 00:54:41,740 --> 00:54:49,040 No assumption on C except. So she could be both positive or negative. 314 00:54:49,040 --> 00:55:00,060 Except in this for me, it looks like it can't be zero. Right here, I got an exponential which is never zero. 315 00:55:00,060 --> 00:55:07,660 So this is positive. I drove the absolute value, and the cost is that now see. 316 00:55:07,660 --> 00:55:13,120 Can also be negative. But somehow zero is missing. 317 00:55:13,120 --> 00:55:21,460 How is that possible? That doesn't look like solid mathematics. 318 00:55:21,460 --> 00:55:27,370 Yes. That's right. 319 00:55:27,370 --> 00:55:36,260 OK, so where did I lose that case? Right here. 320 00:55:36,260 --> 00:55:43,390 So I divide it by white squared minus one. I have I did that. 321 00:55:43,390 --> 00:55:50,030 Ignoring the case when y squared minus one is zero. 322 00:55:50,030 --> 00:55:58,680 So note, so let's call this story here, so in star. 323 00:55:58,680 --> 00:56:04,180 Why minus one is why squared minus one is not zero. 324 00:56:04,180 --> 00:56:14,070 But if we need to allow that because it is possible for white it to be plus or minus one, for example. 325 00:56:14,070 --> 00:56:19,210 If why is the cost and function one, then this is zero. 326 00:56:19,210 --> 00:56:32,630 The idea is zero, so that's OK. So if we allow, if y is plus plus minus one is included. 327 00:56:32,630 --> 00:56:41,100 In the solution. If we allow. 328 00:56:41,100 --> 00:56:55,450 See, to be zero. You know, in the answer. 329 00:56:55,450 --> 00:57:16,990 So then the bottom line is that the answer is. This implicit equation in Y and X, where a C can be any constant. 330 00:57:16,990 --> 00:57:30,940 Good. So be careful the when when you divide by the functioning, why, as I said, here you can do that if you know that's not zero. 331 00:57:30,940 --> 00:57:36,650 But sometimes you get solutions from it being zero, so you have to be careful there. 332 00:57:36,650 --> 00:58:02,343 All right. That's the end of the first lecture, I'll see you tomorrow for for the second lecture and we'll do more differential equations.