Wronskian proof of Sturm comparison
= Wronskian proof of Sturm comparison
For solutions $\varphi_i''+q_i\varphi_i=0$, the mixed Wronskian
$$
W=\varphi_1'\varphi_2-\varphi_1\varphi_2'
$$
satisfies
$$
W'=(q_2-q_1)\varphi_1\varphi_2.
$$
Its endpoint signs between consecutive zeros of a positive $\varphi_1$ force a zero of $\varphi_2$ unless $q_1=q_2$.