1 00:00:05,071 --> 00:00:07,270 PROFESSOR: Welcome in this recitation. 2 00:00:07,270 --> 00:00:10,240 So we're going to talk about linear systems of equations. 3 00:00:10,240 --> 00:00:14,790 So in the first question, we are given a system of equations. 4 00:00:14,790 --> 00:00:18,290 x dot equals 6x plus 5y. 5 00:00:18,290 --> 00:00:20,660 y dot equals x plus 2y. 6 00:00:20,660 --> 00:00:24,350 We're asked to write this system in matrix form. 7 00:00:24,350 --> 00:00:27,820 The second part asks us to convert a differential equation 8 00:00:27,820 --> 00:00:32,119 of second order, x dot dot plus 8x dot plus 7x equals 9 00:00:32,119 --> 00:00:36,710 to zero into matrix form, basically into system of ODEs, 10 00:00:36,710 --> 00:00:39,300 similar to the first part. 11 00:00:39,300 --> 00:00:40,790 In the third part of the problem, 12 00:00:40,790 --> 00:00:42,540 we're asked to interpret the population 13 00:00:42,540 --> 00:00:47,820 model x dot equals 2x minus 3y, y dot equals x minus y. 14 00:00:47,820 --> 00:00:52,380 So here, x and y are modeling either a prey or predator. 15 00:00:52,380 --> 00:00:55,110 And you're asked to think about the interpretation 16 00:00:55,110 --> 00:00:59,060 of the system to determine which of x or y 17 00:00:59,060 --> 00:01:00,630 is the prey or the predator. 18 00:01:00,630 --> 00:01:02,140 So why don't you take a few minutes? 19 00:01:02,140 --> 00:01:03,570 Think about these three questions, 20 00:01:03,570 --> 00:01:04,528 and I'll be right back. 21 00:01:15,190 --> 00:01:16,900 Welcome back. 22 00:01:16,900 --> 00:01:20,120 So for the first question, basically, we're 23 00:01:20,120 --> 00:01:23,460 asked to write this system in matrix form. 24 00:01:23,460 --> 00:01:29,110 So we have [x, y] derivative for that left-hand side. 25 00:01:29,110 --> 00:01:33,950 You need to write this in the form of a matrix multiplying 26 00:01:33,950 --> 00:01:34,920 x and y. 27 00:01:34,920 --> 00:01:42,550 So here, we would have 6, 5; 1, 2. 28 00:01:42,550 --> 00:01:45,360 And that would be our system of differential equations 29 00:01:45,360 --> 00:01:46,760 in matrix form. 30 00:01:46,760 --> 00:01:51,010 And what we would be solving for would be the vector [x, y]. 31 00:01:51,010 --> 00:01:54,850 The second part of the problem, we need to do the opposite, 32 00:01:54,850 --> 00:02:02,780 go from the second order differential equation 33 00:02:02,780 --> 00:02:04,450 into matrix form. 34 00:02:04,450 --> 00:02:12,020 So to do that, we introduced a new variable, y equals x dot. 35 00:02:12,020 --> 00:02:15,923 And from that point, we can then write 36 00:02:15,923 --> 00:02:19,490 x dot dot-- so if I'm going to just start with what we know 37 00:02:19,490 --> 00:02:24,570 about the equation, x dot dot equals-- 38 00:02:24,570 --> 00:02:27,540 let me write it in a system first before I do it 39 00:02:27,540 --> 00:02:28,450 in a vector form. 40 00:02:31,240 --> 00:02:37,790 We would write x dot dot equals minus 7x minus 8x dot. 41 00:02:37,790 --> 00:02:41,760 But we introduced a new variable x dot equals 2y. 42 00:02:41,760 --> 00:02:45,630 So we have minus 7x minus 8y. 43 00:02:45,630 --> 00:02:48,310 So now, the other equation we need 44 00:02:48,310 --> 00:02:51,050 is the one that tells us what this y is. 45 00:02:51,050 --> 00:02:55,940 So we have x dot equals to y, which is the new variable 46 00:02:55,940 --> 00:02:57,200 that we introduced here. 47 00:02:57,200 --> 00:03:00,570 And so we go from a second-order differential equation 48 00:03:00,570 --> 00:03:03,740 into a system of two differential equations that we 49 00:03:03,740 --> 00:03:06,760 can write now in vectorial form, in matrix form, 50 00:03:06,760 --> 00:03:15,200 like we did for the first part, as x, x dot which is just y-- 51 00:03:15,200 --> 00:03:20,550 I'm just going to write this like this, 52 00:03:20,550 --> 00:03:27,420 it's just from what we defined-- equals to, again, [x, y], 53 00:03:27,420 --> 00:03:30,330 like we did previously. 54 00:03:30,330 --> 00:03:32,360 And now we have to read off our system 55 00:03:32,360 --> 00:03:35,690 to find the coefficient of this matrix. 56 00:03:35,690 --> 00:03:37,740 So x dot equals to y means that there 57 00:03:37,740 --> 00:03:43,690 is zero coefficient in front of the x, a 1 in front of the y. 58 00:03:43,690 --> 00:03:46,380 x dot dot equals minus 7x. 59 00:03:46,380 --> 00:03:50,440 So we will have a minus 7 multiplying the x and minus 8 60 00:03:50,440 --> 00:03:51,190 multiplying the y. 61 00:03:53,750 --> 00:03:57,500 And so that's how we convert a differential equation, 62 00:03:57,500 --> 00:04:02,070 second order, into the systems of differential equations 63 00:04:02,070 --> 00:04:03,096 in matrix form. 64 00:04:03,096 --> 00:04:04,970 And this matrix would be called, referred to, 65 00:04:04,970 --> 00:04:08,480 the companion matrix of this differential equation. 66 00:04:08,480 --> 00:04:10,350 OK, so that ends the second part. 67 00:04:14,240 --> 00:04:17,970 So now for the third question, we're 68 00:04:17,970 --> 00:04:24,180 asked to interpret this population dynamics 69 00:04:24,180 --> 00:04:25,890 system of equation. 70 00:04:25,890 --> 00:04:34,100 Minus 3y; y dot equals x minus y. 71 00:04:34,100 --> 00:04:37,220 So the question was, we have two species. 72 00:04:37,220 --> 00:04:39,300 Which one is the prey, which one is the predator? 73 00:04:39,300 --> 00:04:42,030 So how do we go about figuring this out? 74 00:04:42,030 --> 00:04:45,940 Let's look at how x dot varies with y 75 00:04:45,940 --> 00:04:48,070 or basically variable x varies with y. 76 00:04:48,070 --> 00:04:50,870 Here, we can see that we have a coefficient that is minus 3. 77 00:04:50,870 --> 00:04:54,190 It is negative, which means that when y increases, 78 00:04:54,190 --> 00:04:58,760 we have a more and more negative x dot, which means 79 00:04:58,760 --> 00:05:00,430 that the value of x goes down. 80 00:05:00,430 --> 00:05:02,540 So as the population y increases, 81 00:05:02,540 --> 00:05:04,670 we have a decrease of population x, 82 00:05:04,670 --> 00:05:07,820 which suggests that y is a predator eating up 83 00:05:07,820 --> 00:05:09,720 population x. 84 00:05:09,720 --> 00:05:14,140 And if you look at the equation for y, we have x minus y. 85 00:05:14,140 --> 00:05:17,430 And here, what we see is that when x increases, 86 00:05:17,430 --> 00:05:20,520 the population y then increases. 87 00:05:20,520 --> 00:05:23,310 So that definitely confirms that y 88 00:05:23,310 --> 00:05:26,410 is our predator that basically increases 89 00:05:26,410 --> 00:05:29,050 by feeding on the population x. 90 00:05:29,050 --> 00:05:35,845 And as it feeds on population x, y increases, 91 00:05:35,845 --> 00:05:38,470 which means that here this term becomes more and more negative, 92 00:05:38,470 --> 00:05:41,310 which means x decreases in turn. 93 00:05:41,310 --> 00:05:43,780 And these two terms could be modeling, for example, 94 00:05:43,780 --> 00:05:46,190 here just the growth of the population, 95 00:05:46,190 --> 00:05:48,140 so birth term of the prey. 96 00:05:48,140 --> 00:05:50,780 And these minus y here could be just modeling, for example, 97 00:05:50,780 --> 00:05:53,340 a death rate of these predators. 98 00:05:53,340 --> 00:06:01,000 And so we have x prey and y predator. 99 00:06:04,480 --> 00:06:07,040 So from this recitation, we learned 100 00:06:07,040 --> 00:06:10,090 how to convert a system of differential equations 101 00:06:10,090 --> 00:06:11,430 to matrix form. 102 00:06:11,430 --> 00:06:14,770 We learned how to convert a second-order differential 103 00:06:14,770 --> 00:06:17,320 equation into also matrix form, or basically 104 00:06:17,320 --> 00:06:19,130 system differential equation, introducing 105 00:06:19,130 --> 00:06:20,740 notion of companion matrix. 106 00:06:20,740 --> 00:06:23,820 And we learned how to interpret a system of differential 107 00:06:23,820 --> 00:06:27,650 equations in terms of what populations could it 108 00:06:27,650 --> 00:06:30,040 be modeling or what dynamics it could be modeling. 109 00:06:30,040 --> 00:06:32,310 So that ends the recitation.