A nonlinear second order differential equation $x^{\prime \prime}=x-x^{3}+2 x x^{\prime}$
a) Translate this into a first order nonlinear system and determine its equilibrium points.
b) Linearize about each equilibrium point and find the eigenvalues and describe the behavior of the equilibrium. For any point with real eigenvalues compute the two eigenvectors.
c) Now use Maple to generate a phase plane diagram of the system. Put in a variety of initial conditions and study what happens. Write a paragraph on how different trajectories of this system behave and how it is related to the nature of the equilibrium points.
I am given this equation $x'' = x − x^3 + 2xx'$ and asked to translate it into a first order nonlinear equation and then to linearize about each equilibrium point.
I am unsure whether my translation was valid nor which two equations to use for the Jacobian.
I let $x_1=x$ and $x_2=x'$. I got $x_2'=x_1-(x_1^3)+2x_1x_2$ and $x_1'=x_2$ and my equilibrium points were $(0,0), (1,0),(-1,0)$.