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.

If $q_1\le q_2$, then between consecutive zeros of a nontrivial solution ofthere is a zero of every nontrivial solution of $y^{\prime \prime }+q_2(x)y=0$, unless the two coefficients agree throughout that interval.

For solutions ${\phi }_i^{\prime \prime }+q_i{\phi }_i=0$, the mixed WronskiansatisfiesIts endpoint signs between consecutive zeros of a positive ${\phi }_1$ force a zero of ${\phi }_2$ unless $q_1=q_2$.