This is a past exam question, that I don't have a solution to:
Consider the equation: $ \ddot{x} + p(t) \dot{x} + q(t)x = 0 $ when $q$ and $p$ are continuous on $\mathbb{R}$, and there is $a \in \mathbb{R}$ such that $ \forall t \in \mathbb{R}, \space p(t) \leq -a < 0 $
Let $x_1(t), x_2(t)$ two nontrivial solutions to the equations, such that: $ \lim_{t \rightarrow +\infty }(x_1(t)^2 + x_2(t)^2 + \dot{x_1}(t)^2 +\dot{x_2}(t)^2) = 0 $ Prove that $x_1$ and $x_2$ are proportional.
I would love a hint where to start on this one. Thanks!