# Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE

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## 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

\(\displaystyle \map W {y_1, y_2}\) | \(=\) | \(\displaystyle y_1 {y_2}' - y_2 {y_1}'\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map {W'} {y_1, y_2}\) | \(=\) | \(\displaystyle \paren {y_1 {y_2}'' + {y_1}' {y_2}'} - \paren {y_2 {y_1}'' + {y_2}' {y_1}'}\) | Product Rule | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y_1 {y_2}'' - y_2 {y_1}''\) |

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

\(\text {(2)}: \quad\) | \(\displaystyle {y_1}'' + \map P x {y_1}' + \map Q x y_1\) | \(=\) | \(\displaystyle 0\) | ||||||||||

\(\text {(3)}: \quad\) | \(\displaystyle {y_2}'' + \map P x {y_2}' + \map Q x y_2\) | \(=\) | \(\displaystyle 0\) | ||||||||||

\(\text {(4)}: \quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle y_2 {y_1}'' + \map P x y_2 {y_1}' + \map Q x y_2 y_1\) | \(=\) | \(\displaystyle 0\) | $(2)$ multiplied by $y_2$ | ||||||||

\(\text {(5)}: \quad\) | \(\displaystyle y_1 {y_2}'' + \map P x y_1 {y_2}' + \map Q x y_1 y_2\) | \(=\) | \(\displaystyle 0\) | $(3)$ multiplied by $y_1$ | |||||||||

\(\text {(6)}: \quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {y_1 {y_2}'' - y_2 {y_1}''} + \map P x \paren {y_1 {y_2}' - y_2 {y_1}'}\) | \(=\) | \(\displaystyle 0\) | $(5)$ subtracted from $(6)$ |

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.

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.15$: The General Solution of the Homogeneous Equation: Lemma $1$