In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations.

Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating.

Sturm separation theorem

If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x0, x1). It is a special case of the Sturm-Picone comparison theorem.

Proof

are linearly independent it follows that the Wronskian and it turns out, by our result from the Wronskian changed. By the Intermediate Value Theorem.

in between, and it was just proved that this is impossible.

References