# Proving that these functions are proportional

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!

-
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
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

$$|W(x_1,x_2)|=|x_1x_2'-x_1'x_2|\le|x_1x_2'|+|x_1'x_2|\le\frac12(x_1^2+(x_2')^2+(x_1')^2x_2^2),$$ 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 $W(x_1,x_2)(t_0)=0$.
The text passage is a discussion about a mathematical proof involving differential equations and the Wronskian of two solutions. It includes a question asking for a hint to start the proof and a response suggesting the use of the Wronskian.
no_question

the passage does not contain a mathematical question that requires a numerical or symbolic answer. instead, it is a request for a hint to solve a proof and a discussion of a method to approach the proof.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Proving that these functions are proportional

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!

-
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
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

$$|W(x_1,x_2)|=|x_1x_2'-x_1'x_2|\le|x_1x_2'|+|x_1'x_2|\le\frac12(x_1^2+(x_2')^2+(x_1')^2x_2^2),$$ 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 $W(x_1,x_2)(t_0)=0$.
user
The text passage is a discussion about a mathematical proof involving differential equations and the Wronskian of two solutions. It includes a question asking for a hint to start the proof and a response suggesting the use of the Wronskian.

TYPE: NO_QUESTION

The passage does not contain a mathematical question that requires a numerical or symbolic answer. Instead, it is a request for a hint to solve a proof and a discussion of a method to approach the proof.