Any analysis nerds out there know whether the following is possible?

I give you a multivariate diffeq that looks something like

f = x2 + y fxx + fxy + (x + y)f2x

with the stuff on the right being (at worst quadratic) polynomials with positive integer coefficients, maybe multiplied by a copy of the function f you're interested in, maybe squared, and hit with one or two derivatives.

Suppose for the sake of argument that I've given enough boundary conditions to make the solution unique. I then want to know if some mixed repeated derivative, e.g. fxxxyyz is identically zero or not. It would be good enough to find out whether that same derivative is zero or not at a point a little ways away from the origin.

Is this sort of problem ever computationally decidable, perhaps by means of numerical approximation? It sounds pretty nasty to me, but I am no expert in analysis, and Doron Zeilberger's recent April Fool's joke was inspiring in its generatingfunctionological lunacy.

If the above problem is, then I think MELL (a subset of linear logic whose decidability is an open problem) is decidable. If you can do it without the quadratic appearance of fs, then at least you get an alternate proof of the (known) decidability of Petri Net reachability.
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