I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion.

Consider a classical mechanical system with one degree on freedom $q$ and a Hamiltonian $H(q,p,t)$. The equations of motion can be cast as the Euler-Lagrange equations which are second-order ODE, non-linear in the generic case. Yet another equivalent way is through the Hamilton-Jacobi equation $$ \frac{\partial S}{\partial t}+H\left(q,\frac{\partial S}{\partial q},t\right)=0 \qquad (1)$$ which is a single non-linear PDE.

Quantum-mechanically this system is described by a linear Schrodinger equation (I gloss over all ambiguities and subtleties that might be in quantizing a classical system) $$-i\hbar\frac{\partial \psi}{\partial t}+H(q,-i\hbar\partial_q,t)\psi=0 \qquad (2)$$

Formally, equation (1) is the $\hbar\to0$ limit of equation (2) if one makes the following ansatz $\psi=e^{iS/\hbar}$. So it seems that at least formally for a large class of ODE one can find an equivalent linear PDE. I am interested to learn if there is any systematic theory to back up this heuristic arguments.