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 $(1)$:
\((2):\quad\) | \(\displaystyle {y_1}'' + \map P x {y_1}' + \map Q x y_1\) | \(=\) | \(\displaystyle 0\) | ||||||||||
\((3):\quad\) | \(\displaystyle {y_2}'' + \map P x {y_2}' + \map Q x y_2\) | \(=\) | \(\displaystyle 0\) | ||||||||||
\((4):\quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle y_2 {y_1}'' + \map P x y_2 {y_1}' + \map Q x y_2 y_1\) | \(=\) | \(\displaystyle 0\) | $(2)$ multiplied by $y_2$ | ||||||||
\((5):\quad\) | \(\displaystyle y_1 {y_2}'' + \map P x y_1 {y_2}' + \map Q x y_1 y_2\) | \(=\) | \(\displaystyle 0\) | $(3)$ multiplied by $y_1$ | |||||||||
\((6):\quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {y_1 {y_2}'' - y_2 {y_1}''} + \map P x \paren {y_1 {y_2}' - y_2 {y_1}'}\) | \(=\) | \(\displaystyle 0\) | $(5)$ 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 $1$