The Wronskian: all or nothing, by Abel's identity

Ordinary Differential EquationsODE, PDE & Applied Mathematicsfree

Why a Wronskian that vanishes once vanishes always — and the hypothesis that makes it fail.

Step 1 / 4Abel's identity

For $y^{''} + p (x) y^{'} + q (x) y = 0$ with $p, q$ continuous on an interval $I$ , the Wronskian of two solutions satisfies $W^{'} = - p W$ , so

$W (x) = W (x_{0}) exp (- \int_{x_{0}}^{x} p) .$

The exponential never vanishes.