Solutions of Linear 2nd Order ODE have Common Zero iff Linearly Dependent

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 $y_2$ both have a zero for the same value of $x$ in $\left[{a \,.\,.\, b}\right]$.

Then $y_1$ and $y_2$ are constant multiples of each other.

That is, $y_1$ and $y_2$ are linearly dependent.

Proof
Let $\xi \in \left[{a \,.\,.\, b}\right]$ be such that $y_1 \left({\xi}\right) = y_2 \left({\xi}\right) = 0$.

Consider the Wronskian $W \left({y_1, y_2}\right)$ at $\xi$:

From Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE:
 * $\forall x \in \left[{a \,.\,.\, b}\right]: W \left({y_1 \left({x}\right), y_2 \left({x}\right)}\right) = 0$

and so from Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent:
 * $y_1$ and $y_2$ are linearly dependent.