How do I find nullclines for an ODE given only one second order ODE?

Question

My textbook asks me to draw an outline of the phase space for $\frac{d^2x}{dt^2}-x\frac{dx}{dt}+x^2=0$ by studying the nullclines, but it never showed how to find the nullclines for a single equation like this, only with a system of equations. How can I accomplish this?

Answer 1

Hint

Write your equation with a system...

$$\frac{d^2x}{dt^2}-x\frac{dx}{dt}+x^2=0$$ $$ \implies \begin{cases} x'=y \\ y'=xy-x^2 \end{cases} $$