In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain.
Let pi, qi for i = 1, 2 be real-valued continuous functions on the interval [a, b] and let be two homogeneous linear second order differential equations in self-adjoint form with and Let u be a non-trivial solution of (1) with successive roots at z1 and z2 and let v be a non-trivial solution of (2).
Then one of the following properties holds.
The first part of the conclusion is due to Sturm (1836),[1] while the second (alternative) part of the theorem is due to Picone (1910)[2][3] whose simple proof was given using his now famous Picone identity.
In the special case where both equations are identical one obtains the Sturm separation theorem.