Let $a,b \in \mathbb{R}$ with $a < b$. Let $P \in C^1([a,b];\mathbb{R})$ and $Q \in C([a,b];\mathbb{R})$, and assume $P(x)>0$ for every $x \in [a,b]$. Let $u,v \in C^1([a,b];\mathbb{R})$ be linearly independent over $\mathbb{R}$ solutions of the Sturm-Liouville equation

\begin{align*} -(P w')' + Qw = 0 \end{align*}

on $[a,b]$, meaning that $P u', P v' \in C^1([a,b];\mathbb{R})$ and both $u$ and $v$ satisfy the displayed equation pointwise on $[a,b]$. If $\alpha,\beta \in [a,b]$ with $\alpha < \beta$ are consecutive zeros of $u$, in the sense that $u(\alpha)=u(\beta)=0$ and $u(x)\neq 0$ for every $x \in (\alpha,\beta)$, then there exists a unique $\gamma \in (\alpha,\beta)$ such that $v(\gamma)=0$.