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

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

such that $y_1$ is not the trivial solution.

Let $\map {y_2} x$ be the real function defined as:
 * $\map {y_2} x = \map v x \, \map {y_1} x$

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

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 $v$ $x$:


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

As $\displaystyle -\int P \rd x$ is a real function, $e^{-\int P \rd x}$ 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.