Let $M$ be a Riemannian manifold and $h, f: M \to \mathbb{R}$ be smooth functions, and $h$ is positive. Does there always exist another smooth function $\psi: M \to \mathbb{R} $ (or, at least, $\psi: U \to \mathbb{R}$ where $U$ is an open subset of $M$)such that $$\nabla^2 \psi = h \ \nabla^2 f$$
I think the answer will be yes because really we're just going to have to "integrate locally", but I'm not sure that can be formalized. I'd appreciate any help.