P6. Consider the differential equation x”

P6. Consider the differential equation $x'' - ax'(x^2 + x + 1) + x = 0$ where $a$ is a constant. (a) Rewrite this equation as an equivalent system of first order differential equations. (b) Find the equilibrium solution (there is only one!) for the system. (c) Determine all the possible values of the unspecified parameter $a$ for which the equilibrium solution is asymptotically stable.

Transcript

00:03 Suppose we have the differential equation y prime of t is equal to 6 minus 2 y.

00:09 Now, if we were to try to find our equilibrium points when plotting our directional field, what we do is set it equal to 0 and solve for y.

00:31 This tells us that when y is equal to 3, our slope is flat or 0, which would give us a horizontal line.

00:42 And this point, or this line, will be our.

00:45 Equilibrium because if y ever equals three at any point in time, it will continue to stay three for as long as the function is running.

01:00 So now to find our slopes on other points in our in our differential equation, what we do is we'll start by choosing four and we'll plot that into our differential equation.

01:17 6 minus 2 times 4 for y, because we're going to be working on that point.

01:23 And that is equal to 6 minus 8, which is equal to negative 2.

01:31 So this tells us that the slope, whenever y is 4, or whenever y is 4, our slope is negative 2.

01:40 So it faces down, and it has a slope of negative 2.

01:48 Or it has a slope of two facing down, which makes it negative.

01:54 Now to find our other values, we do the same thing, but just change whatever we input as y.

02:03 So let's go to 2, so 2 times 2 is equal to 6 minus 4, which is equal to positive 2.

02:11 So if it's positive 2, it'll just, it'll have the same magnitude as when y was equal to 4, but it will be running in the opposite direction...

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