Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE

Theorem
Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE:
 * $(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$

on a closed interval $\closedint a b$.

Let $y_1$ and $y_2$ be linearly independent.

Then their Wronskian is either never zero, or zero everywhere on $\closedint a b$.

Proof
Because $y_1$ and $y_2$ are both particular solutions of $(1)$:

That is:


 * $\dfrac {\d P} {\d W} + P W = 0$

This is a linear first order ODE.

From Solution to Linear First Order Ordinary Differential Equation:
 * $W = C e^{-\int P \rd x}$

The exponential function is never zero:

Therefore:
 * $W = 0 \iff C = 0$

and the result follows.