Find the critical points of the following second-order equation (by transferring into a system of first order differential equations) and determine the stability of these critical points:

My start at a solution: transferring into a system of first order differential equations

let

therefore

So to get the critical points I can set This implies

so , where

so , implies

Then my critical points are

(1,0) and (-1,0)

At this point I would take the Jacobian at the two critical points and solve for the eigenvalues. The eigenvalues would then tell me about the stability? I will do this, I just want to confirm I am on the right track first.

Thanks so much,

Len