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Not sure if you still need an answer, but here goes;

Define g(x) = f(x) - x*y, this makes the given equation homogeneous;

d^2/dx^2 d/dy g(x, y) = g(x, y)

And now assume g(x, y) = A(x)B(y). That lets you factorize the equation into two parts;

d^2/xx^2 A(x) = a A(x), d/dy B(y) = (1/a) B(y), with some positive constant a.

This can be solved as A(x) = C1 exp(+-sqrt(a) x), B(y) = C2 exp(y / a)

And by multiplying them back,

g(x, y) = (C1 * C2) exp(+-sqrt(a) x + y / a) = C exp(+-sqrt(a) x + y / a)

Therefore, f(x, y) = C exp(+-sqrt(a) x + y / a) + x*y.

General solution is linear combination of these solution with different values of C and a.

But I might have missed something there, because this solution contains only two arbitrary parameters whereas the original equation is a third order differential equation.