The problem is stated as follows:

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a) Transform the differential equation

with the substitution and in the region y>0.

b) Determine all solutions of class (that is, solutions with continuous second (partial) derivatives) for the differential equation in this region. The solutions should be expressed as functions of the variables x and y.

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My question is about part b), since I don't know how I should get started on that. I'll show some of my work on part a) for some context:

(since y>0).

Also, by the looks of the PDE, we have , so we can use the chain rule:

By finding the partials of x and y with respect to u and v, we get

.

Differentiating again, and substituting the relevant partials back into the original PDE, we eventually, after a few pages of work, get

(assuming I didn't mess up somewhere).

I'm not sure what to make of that, other than how we have a function that depends on u and v, and consequently, on x and y. But, I guess that was kind of the premise for the problem. How do I approach part b)?

I suppose I could integrate with respect to one variable at a time and get or (C and D being integration constants), but I suspect that won't cover all possible cases.