Skip to content
Interactive Explainer

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.

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

Step 1 / 4Abel's identity

For y′′+p(x)y′+q(x)y=0y'' + p(x)y' + q(x)y = 0 with p,qp,q continuous on an interval II, the Wronskian of two solutions satisfies W′=−pWW' = -pW, so

W(x)=W(x0)exp⁡(−∫x0xp).W(x) = W(x_0)\exp\left(-\int_{x_0}^x p\right).

The exponential never vanishes.

ShareWhatsAppTelegram