A system of differential equations is given. (a) Construct the phase plane, plotting all nullcli

step 1 Okay, so let's start this by finding the null clines. So for the x null cline, we find that by s

A system of differential equations is given. (a) Construct...

Key Concepts

Nullclines

Nullclines are curves in the phase plane where one of the derivatives in the system vanishes; that is, they are the loci where either x' or y' equals zero. These curves help in decomposing the dynamics by indicating where the rate of change in one direction is zero, facilitating the location of equilibrium points as their intersections.

System of Differential Equations

This concept involves a set of equations that describe how multiple variables evolve with respect to an independent variable, typically time. These systems are used to model interdependent processes and their interaction dynamics, and a qualitative analysis can provide insight into the behavior of the system over time.

Phase Plane Analysis

Phase plane analysis is a technique used to study two-dimensional systems of differential equations by plotting the trajectories in a plane where each point represents a state of the system. By examining these trajectories, one can understand the system's dynamics, identify patterns, and predict long-term behavior.

Equilibrium Points

Equilibrium points, or steady states, are points in the phase plane where the time derivatives of all variables are zero, meaning the system experiences no change at these points. Analyzing equilibria is crucial for understanding the stability and long-term behavior of the system, as small perturbations near these points can indicate whether the system returns to equilibrium or diverges away.

Transcript

00:01 Okay, so let's start this by finding the null clines.

00:03 So for the x null cline, we find that by setting this equal to zero, the derivative x prime equal to zero.

00:15 Now we have two factors.

00:16 We have this here and then this here as well.

00:21 So we're going to set x minus two equal to zero.

00:25 So we get an null client of x equals two.

00:27 Or we can also set ln of xy equal to 0.

00:31 Now ln of xy is only equal to 0 if xy is equal to 1 or so we get y equals 1 over x.

00:42 Now for the next, for the y no climbs, we set y prime equal to 0.

00:48 I'll get e to the x times x minus y equals 0.

00:52 E to the x can never equal 0.

00:55 So we can ignore that.

00:56 So we only have x minus y equals zero or y is equal to x.

01:02 Now, we're only graphing in the region x comma y strictly greater than zero.

01:09 Okay, so now let's graph our region then, the first quadrant.

01:15 Now we have y or x equals two.

01:17 So that's a vertical line at two, right? i'm just going to call that two.

01:24 And then one over x.

01:25 So like so.

01:28 Okay.

01:29 Now we also have the line y equals x.

01:31 So that's going to look like so.

01:37 So now we need to find the equilibrium points.

01:43 So we need to set the this here, this function equal to this function and this function.

01:55 For this one here, this intersection here, this is obviously going to be 2 comma 2.

02:02 Now to find this intersection, we need to set x equal to 1 over x.

02:09 So when is x equal to 1 over x? so x is equal to 1 over x, right? x squared is equal to 1 or x is equal to plus or minus 1.

02:21 But since we only care about the positive one, right, we can ignore the negative part.

02:28 Right? so we can just do plus 1.

02:31 Oh, sorry, plus one.

02:36 So then this equilibrium point is going to be 1 comma 1, like so.

02:41 Now for the regions, right, we need to find some test points as well.

02:51 So let's find, so below y equals x, we can use a test point, say, 1, 3, for example.

03:09 1 -3 or 3 -1 will be about here.

03:13 Okay? so let's do the test point 3 comma 1.

03:18 If we do 3 comma 1, so negative 3 minus 1 times ln of 3.

03:28 3, well, 3 times 1, which is ln of 3.

03:30 So we get negative 2 ln of 3...

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