WEBVTT
00:00:05.071 --> 00:00:07.270
PROFESSOR: Welcome
in this recitation.
00:00:07.270 --> 00:00:10.240
So we're going to talk about
linear systems of equations.
00:00:10.240 --> 00:00:14.790
So in the first question, we
are given a system of equations.
00:00:14.790 --> 00:00:18.290
x dot equals 6x plus 5y.
00:00:18.290 --> 00:00:20.660
y dot equals x plus 2y.
00:00:20.660 --> 00:00:24.350
We're asked to write this
system in matrix form.
00:00:24.350 --> 00:00:27.820
The second part asks us to
convert a differential equation
00:00:27.820 --> 00:00:32.119
of second order, x dot dot
plus 8x dot plus 7x equals
00:00:32.119 --> 00:00:36.710
to zero into matrix form,
basically into system of ODEs,
00:00:36.710 --> 00:00:39.300
similar to the first part.
00:00:39.300 --> 00:00:40.790
In the third part
of the problem,
00:00:40.790 --> 00:00:42.540
we're asked to
interpret the population
00:00:42.540 --> 00:00:47.820
model x dot equals 2x minus
3y, y dot equals x minus y.
00:00:47.820 --> 00:00:52.380
So here, x and y are modeling
either a prey or predator.
00:00:52.380 --> 00:00:55.110
And you're asked to think
about the interpretation
00:00:55.110 --> 00:00:59.060
of the system to
determine which of x or y
00:00:59.060 --> 00:01:00.630
is the prey or the predator.
00:01:00.630 --> 00:01:02.140
So why don't you
take a few minutes?
00:01:02.140 --> 00:01:03.570
Think about these
three questions,
00:01:03.570 --> 00:01:04.528
and I'll be right back.
00:01:15.190 --> 00:01:16.900
Welcome back.
00:01:16.900 --> 00:01:20.120
So for the first
question, basically, we're
00:01:20.120 --> 00:01:23.460
asked to write this
system in matrix form.
00:01:23.460 --> 00:01:29.110
So we have [x, y] derivative
for that left-hand side.
00:01:29.110 --> 00:01:33.950
You need to write this in the
form of a matrix multiplying
00:01:33.950 --> 00:01:34.920
x and y.
00:01:34.920 --> 00:01:42.550
So here, we would
have 6, 5; 1, 2.
00:01:42.550 --> 00:01:45.360
And that would be our system
of differential equations
00:01:45.360 --> 00:01:46.760
in matrix form.
00:01:46.760 --> 00:01:51.010
And what we would be solving
for would be the vector [x, y].
00:01:51.010 --> 00:01:54.850
The second part of the problem,
we need to do the opposite,
00:01:54.850 --> 00:02:02.780
go from the second order
differential equation
00:02:02.780 --> 00:02:04.450
into matrix form.
00:02:04.450 --> 00:02:12.020
So to do that, we introduced a
new variable, y equals x dot.
00:02:12.020 --> 00:02:15.923
And from that point,
we can then write
00:02:15.923 --> 00:02:19.490
x dot dot-- so if I'm going to
just start with what we know
00:02:19.490 --> 00:02:24.570
about the equation,
x dot dot equals--
00:02:24.570 --> 00:02:27.540
let me write it in a
system first before I do it
00:02:27.540 --> 00:02:28.450
in a vector form.
00:02:31.240 --> 00:02:37.790
We would write x dot dot
equals minus 7x minus 8x dot.
00:02:37.790 --> 00:02:41.760
But we introduced a new
variable x dot equals 2y.
00:02:41.760 --> 00:02:45.630
So we have minus 7x minus 8y.
00:02:45.630 --> 00:02:48.310
So now, the other
equation we need
00:02:48.310 --> 00:02:51.050
is the one that tells
us what this y is.
00:02:51.050 --> 00:02:55.940
So we have x dot equals to
y, which is the new variable
00:02:55.940 --> 00:02:57.200
that we introduced here.
00:02:57.200 --> 00:03:00.570
And so we go from a second-order
differential equation
00:03:00.570 --> 00:03:03.740
into a system of two
differential equations that we
00:03:03.740 --> 00:03:06.760
can write now in vectorial
form, in matrix form,
00:03:06.760 --> 00:03:15.200
like we did for the first part,
as x, x dot which is just y--
00:03:15.200 --> 00:03:20.550
I'm just going to
write this like this,
00:03:20.550 --> 00:03:27.420
it's just from what we defined--
equals to, again, [x, y],
00:03:27.420 --> 00:03:30.330
like we did previously.
00:03:30.330 --> 00:03:32.360
And now we have to
read off our system
00:03:32.360 --> 00:03:35.690
to find the coefficient
of this matrix.
00:03:35.690 --> 00:03:37.740
So x dot equals to
y means that there
00:03:37.740 --> 00:03:43.690
is zero coefficient in front of
the x, a 1 in front of the y.
00:03:43.690 --> 00:03:46.380
x dot dot equals minus 7x.
00:03:46.380 --> 00:03:50.440
So we will have a minus 7
multiplying the x and minus 8
00:03:50.440 --> 00:03:51.190
multiplying the y.
00:03:53.750 --> 00:03:57.500
And so that's how we convert
a differential equation,
00:03:57.500 --> 00:04:02.070
second order, into the systems
of differential equations
00:04:02.070 --> 00:04:03.096
in matrix form.
00:04:03.096 --> 00:04:04.970
And this matrix would
be called, referred to,
00:04:04.970 --> 00:04:08.480
the companion matrix of
this differential equation.
00:04:08.480 --> 00:04:10.350
OK, so that ends
the second part.
00:04:14.240 --> 00:04:17.970
So now for the third
question, we're
00:04:17.970 --> 00:04:24.180
asked to interpret this
population dynamics
00:04:24.180 --> 00:04:25.890
system of equation.
00:04:25.890 --> 00:04:34.100
Minus 3y; y dot
equals x minus y.
00:04:34.100 --> 00:04:37.220
So the question was,
we have two species.
00:04:37.220 --> 00:04:39.300
Which one is the prey,
which one is the predator?
00:04:39.300 --> 00:04:42.030
So how do we go about
figuring this out?
00:04:42.030 --> 00:04:45.940
Let's look at how
x dot varies with y
00:04:45.940 --> 00:04:48.070
or basically variable
x varies with y.
00:04:48.070 --> 00:04:50.870
Here, we can see that we have
a coefficient that is minus 3.
00:04:50.870 --> 00:04:54.190
It is negative, which means
that when y increases,
00:04:54.190 --> 00:04:58.760
we have a more and more
negative x dot, which means
00:04:58.760 --> 00:05:00.430
that the value of x goes down.
00:05:00.430 --> 00:05:02.540
So as the population
y increases,
00:05:02.540 --> 00:05:04.670
we have a decrease
of population x,
00:05:04.670 --> 00:05:07.820
which suggests that y
is a predator eating up
00:05:07.820 --> 00:05:09.720
population x.
00:05:09.720 --> 00:05:14.140
And if you look at the equation
for y, we have x minus y.
00:05:14.140 --> 00:05:17.430
And here, what we see is
that when x increases,
00:05:17.430 --> 00:05:20.520
the population y then increases.
00:05:20.520 --> 00:05:23.310
So that definitely
confirms that y
00:05:23.310 --> 00:05:26.410
is our predator that
basically increases
00:05:26.410 --> 00:05:29.050
by feeding on the population x.
00:05:29.050 --> 00:05:35.845
And as it feeds on
population x, y increases,
00:05:35.845 --> 00:05:38.470
which means that here this term
becomes more and more negative,
00:05:38.470 --> 00:05:41.310
which means x decreases in turn.
00:05:41.310 --> 00:05:43.780
And these two terms could
be modeling, for example,
00:05:43.780 --> 00:05:46.190
here just the growth
of the population,
00:05:46.190 --> 00:05:48.140
so birth term of the prey.
00:05:48.140 --> 00:05:50.780
And these minus y here could
be just modeling, for example,
00:05:50.780 --> 00:05:53.340
a death rate of these predators.
00:05:53.340 --> 00:06:01.000
And so we have x
prey and y predator.
00:06:04.480 --> 00:06:07.040
So from this
recitation, we learned
00:06:07.040 --> 00:06:10.090
how to convert a system
of differential equations
00:06:10.090 --> 00:06:11.430
to matrix form.
00:06:11.430 --> 00:06:14.770
We learned how to convert
a second-order differential
00:06:14.770 --> 00:06:17.320
equation into also
matrix form, or basically
00:06:17.320 --> 00:06:19.130
system differential
equation, introducing
00:06:19.130 --> 00:06:20.740
notion of companion matrix.
00:06:20.740 --> 00:06:23.820
And we learned how to interpret
a system of differential
00:06:23.820 --> 00:06:27.650
equations in terms of
what populations could it
00:06:27.650 --> 00:06:30.040
be modeling or what dynamics
it could be modeling.
00:06:30.040 --> 00:06:32.310
So that ends the recitation.