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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!

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Have you tried looking at the Wronskian? $x_1, x_2$ are proportional exactly when their Wronskian is zero, and their Wronskian is constant. –  Giuseppe Negro Sep 1 '12 at 17:28
    
@GiuseppeNegro I thought about it, but I couldn't come up with anything intelligent to say about it based on the assumptions in the question. –  Hila Sep 1 '12 at 17:41
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I'm sorry, I erroneously wrote that the Wronskian is constant. This is not the case: truth is that the Wronskian is strictly increasing, and this is the property that Julian Aguirre employs below. –  Giuseppe Negro Sep 1 '12 at 18:35
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1 Answer

up vote 4 down vote accepted

$$ |W(x_1,x_2)|=|x_1x_2'-x_1'x_2|\le|x_1x_2'|+|x_1'x_2|\le\frac12(x_1^2+(x_2')^2+(x_1')^2x_2^2), $$ so that $\lim_{t\to\infty}W(x_1,x_2)(t)=0$.

On the other hand, for any $t_0\in\mathbb{R}$ $$ |W(x_1,x_2)(t)|=|W(x_1,x_2)(t_0)|e^{-\int_0^tp(s)ds}\ge |W(x_1,x_2)(t_0)|e^{at}\quad\forall t>t_0. $$ It follows that $W(x_1,x_2)(t_0)=0$.

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