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

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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 $y_2$ 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 $\xi$:

\(\displaystyle \map W {\map {y_1} \xi, \map {y_2} \xi}\) \(=\) \(\displaystyle \map {y_1} \xi \map { {y_2}'} \xi - \map {y_2} \xi \map { {y_1}'} \xi\)
\(\displaystyle \) \(=\) \(\displaystyle 0 \cdot \map { {y_2}'} \xi - 0 \cdot \map { {y_1}'} \xi\)
\(\displaystyle \) \(=\) \(\displaystyle 0\)


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