 Sturm–Picone comparison theorem

For two real self-adjoint equations $(p_i{y}_i\prime )\prime +q_i{y}_i=0$ with continuous coefficients, $0<p_2\le p_1$ and $q_1\le q_2$, between consecutive zeros of a nontrivial first solution a nontrivial second solution either has a zero or is proportional to the first there. This comparison controls oscillation through the coefficients.

The Sturm-Picone comparison theorem is a fundamental result in the theory of ordinary differential equations, especially in the context of second-order linear differential equations.