Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE

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

on a closed interval $\left[{a \,.\,.\, b}\right]$.

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

Then their Wronskian is either never zero, or zero everywhere on $\left[{a \,.\,.\, b}\right]$.

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

That is:


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

This is a linear first order ODE.

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

The exponential function is never zero,

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

and the result follows.