Solve Linear Systems: Determine Critical Point Stability

[forest125]

Wanted to brush up on my diff equations a bit. I have kind of forgotten how to determine phase plane stability in systems of differential equations.

Homework Statement

Solve linear systems. Determine whether the critical point (0,0) is stable, asymptotically stable, or unstable.

dx/dt=2x and dy/dt=-2y

Homework Equations

Not sure there's anything for this section...

The Attempt at a Solution

Well 2x=0 and -2y=0 only yields a 1 CP of (0,0). Without using MATLAB or something, I'm really not sure how to do this numerically. Help anyone?

Thank you much.

 

[HallsofIvy]

Do what numerically? Surely, you didn't need MatLab to solve 2x= 0 and -y= 0!

If fact, it is not difficult to solve the two equations separately. If dx/dt= 2x, then dx/x= 2dt and, integrating, ln(x)= 2ln(t)+ C= ln(t^2)+ C so that x= C't^2. Obviously, as t goes to infinity, so does x so this is an unstable equilibrium.

More generally, if you have dx/dt= ax+ by, dy/dt= cx+ dy, you would look at the eigenvalues of the matrix
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$

If the real part of any of the eigenvalues is positive, the equilibrium is unstable. If all eigenvalues have negative real part, the equilibrium is asymptotically stable, if any of the eigenvalues have 0 real part and the others negative, the equilibrium is stable.