For the general question I vote no. As you formulated I see it, potentially, as a question of differential algebra. Although there is no definition of physical proccess I imagine it can be formalised through constraints on the field extensions allowed to solve your equation. The prototypical result I have in mind is Liouville's Theorem.
If instead you specialize the general question to Painleve's equation then I would bet that the answer is yes. Painlevé equation is one of these ubiquitous objects in Mathematics and I would not be surprised if it models a physical phenomena. I already crossed with a Springer Lecture Notes relating it to the geometry of surfaces.
As a side remark let me notice that Painlevé's equations were originally found not as equations governing isomonodromic deformations but instead as non-linear second order equations having the so called Painlevé property (absence of movable singular points).
