Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE 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 $\left[{a \,.\,.\, b}\right]$.

Then:
 * $y_1$ and $y_2$ are linearly dependent


 * the Wronskian $W \left({y_1, y_2}\right)$ of $y_1$ and $y_2$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.
 * the Wronskian $W \left({y_1, y_2}\right)$ of $y_1$ and $y_2$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.

Sufficient Condition
Let $y_1$ and $y_2$ are linearly dependent.

Suppose either $y_1$ or $y_2$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.

Then either ${y_1}'$ or ${y_2}'$ is also zero everywhere on $\left[{a \,.\,.\, b}\right]$.

Thus:
 * $y_1 {y_2}' - y_2 {y_1}' = 0$

and so by definition $W \left({y_1, y_2}\right) = 0$ everywhere on $\left[{a \,.\,.\, b}\right]$.

Suppose that neither $y_1$ nor $y_2$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.

, it follows by definition of linearly dependent that:
 * $y_2 = C y_1$

for some $C \in \R$.

Thus by Derivative of Constant Multiple:
 * ${y_2}' = C {y_1}'$

Hence:

Hence by definition $W \left({y_1, y_2}\right) = 0$ everywhere on $\left[{a \,.\,.\, b}\right]$.

Necessary Condition
Let $W \left({y_1, y_2}\right) = 0$ everywhere on $\left[{a \,.\,.\, b}\right]$.

One way for this to happen is for either $y_1$ or $y_2$ to be zero everywhere on $\left[{a \,.\,.\, b}\right]$.

, suppose $y_1$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.

Then by Real Function is Linearly Dependent with Zero Function, $y_1$ and $y_2$ are linearly dependent.

Suppose that neither $y_1$ nor $y_2$ is zero everywhere on $\left[{a \,.\,.\, b}\right]$.

As $y_1$ is continuous, there is some closed interval:
 * $\left[{c \,.\,.\, d}\right] \subseteq \left[{a \,.\,.\, b}\right]$

on which:
 * $\forall x \in \left[{c \,.\,.\, d}\right]: y_1 \left({x}\right) \ne 0$

Since $W \left({y_1, y_2}\right) = 0$:
 * $\dfrac {y_1 {y_2}' - y_2 {y_1}'} { {y_1}^2} = 0$

on $\left[{c \,.\,.\, d}\right]$.

By Quotient Rule for Derivatives, this is the same as:
 * $\left({\dfrac {y_2} {y_1} }\right)' = 0$

Integrating $x$, this gives:
 * $\dfrac {y_2} {y_1} = k$

Explicitly:
 * $\forall x \in \left[{c \,.\,.\, d}\right]: y_2 \left({x}\right) = k y_1 \left({x}\right)$

As $y_2 \left({x}\right)$ and $k y_1 \left({x}\right)$ have equal values on $\left[{c \,.\,.\, d}\right]$, their derivatives are equal there as well.

From Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions, it follows that:
 * $\forall x \in \left[{a \,.\,.\, b}\right]: y_2 \left({x}\right) = k y_1 \left({x}\right)$.