Induced Solution to Homogeneous Linear Second Order ODE is Linearly Independent with Inducing Solution

Theorem
Let $y_1 \left({x}\right)$ be a particular solution to the homogeneous linear second order ODE:
 * $(1): \quad \dfrac {\mathrm d^2 y} {\mathrm d x^2} + P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} + Q \left({x}\right) y = 0$

such that $y_1$ is not the trivial solution.

Let $y_2 \left({x}\right)$ be the real function defined as:
 * $y_2 \left({x}\right) = v \left({x}\right) y_1 \left({x}\right)$

where:
 * $\displaystyle v = \int \dfrac 1 { {y_1}^2} e^{- \int P \, \mathrm d x} \, \mathrm d x$

Then $y_2$ and $y_1$ are linearly independent.

Proof
This will be demonstrated by calculating the Wronskian of $y_1$ and $y_2$ and demonstrating that it is non-zero everywhere.

First we take the derivative of $v$ $x$:


 * $v' = \dfrac 1 { {y_1}^2} e^{- \int P \, \mathrm d x}$

As $\displaystyle - \int P \, \mathrm d x$ is a real function, $e^{- \int P \, \mathrm d x}$ is non-zero wherever it is defined.

Hence from Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent, $y_1$ and $y_2$ are linearly independent.