Sturm separation theorem

The zeros of two linearly independent solutions of the Airy equation $y''-xy=0$ alternate, as predicted by the Sturm separation theorem.

In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solution of such an equations the roots of the two solutions are alternating.

Sturm separation theorem

Given a homogeneous second order linear differential equation and two linear independent solutions u(x) and v(x) with x₀ and x₁ successive roots of u(x), then v(x) has exactly one root in the open interval ]x₀, x₁[. This can be proved by applying Rolle's theorem to the quotient u(x)/v(x); a contradiction would appear unless v(x)=0 somewhere in the open interval ]x₀, x₁[.