If p(0,0) and h(0) are positive, prove that the origin is an asymptotically stable critical point for the dynamical equation:
x'' + p(x,x')x' + xh(x) = 0.
I tried setting y = x' and y' = x'' and setting up a first order system of two equations to find the eigenvalues and determining the stability of the system based on that, but I ended up with a really ugly Jacobian and don't think I'm on the right path. I don't know how else I could possibly solve it.