Given the system of ODE: $\ x' = y$, $\ y' = -f(x)$ where $f: R\rightarrow R$ is Lipschitz continuous. Let $F: R\rightarrow R$ be any antiderivative of $f$, and let $\ H(x,y) = \frac{y^2}{2} + F(x)$. Describe how to find the phase portraits of this nonlinear system, given that for any solution $\alpha: I\rightarrow R$, $H(\alpha(t)) = C$ $\ \forall\ t\in I$.
My attempt: I was able to show that for any solution $\alpha: I\rightarrow R$, $H(\alpha(t)) = C$ $\ \forall\ t\in I$. Then I considered the level set of the solutions $(x(t), y(t))$ such that $H(x,y) = C$. Differentiating both sides w.r.t $\ t$, I get: $yy' + f = 0$, or $\ f(1-y) = 0$ (since $\ f$ is Lipschitz continuous function). This implies $\ f = 0$ or $\ y(t) = 1$.
Thus, $(x,y) = C(t, 1)$. So the phase portrait is a horizontal line $y=C$ for any constant C.
Can anyone please help verify my solution above, and correct my mistake(s) if any? Any help would greatly be appreciated.