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

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 $y_2$ both have a zero for the same value of $x$ in $\closedint a b$.

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

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

Proof

Let $\xi \in \closedint a b$ be such that $\map {y_1} \xi = \map {y_2} \xi = 0$.

Consider the Wronskian $\map W {y_1, y_2}$ at $\xi$:

\(\ds \map W {\map {y_1} \xi, \map {y_2} \xi}\)

\(=\)

\(\ds \map {y_1} \xi \map { {y_2}'} \xi - \map {y_2} \xi \map { {y_1}'} \xi\)

\(\ds \)

\(=\)

\(\ds 0 \cdot \map { {y_2}'} \xi - 0 \cdot \map { {y_1}'} \xi\)

\(\ds \)

\(=\)

\(\ds 0\)

From Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE:

$\forall x \in \closedint a b: \map W {\map {y_1} \xi, \map {y_2} \xi} = 0$

and so from Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent:

$y_1$ and $y_2$ are linearly dependent.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.14$: Problem $7$