Question

Find each equilibrium solution x(t)=x_0 of the given second-order differential equation x"+f(x,x')=0. Use a computer system or graphing calculator to construct a phase portrait and direction field for the equivalent first-order system x'y, y'=-f(x,y). Thereby ascertain whether the critical point (x_0,0) looks like a center, a saddle point, or a spiral point of this system.

Answer 1

@mathmale

Answer 2

I made a typo. x'=y up in the question not x'y