C2-solutions to a diff.equation.

[Hatmpatn]

I have the following problem:
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Calculate all the C2-solutions z(x,y) to the differential equation:

with the following constraint:

by making the substitution u=xy, v=x

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Solution
I've begun slightly but this doesn't take me far..

 

[pasmith]

Start by working out what the differential operators \frac{\partial}{\partial x} and \frac{\partial}{\partial y} are in terms of u, v and the differential operators \frac{\partial}{\partial u} and \frac{\partial}{\partial v}. From these you can find expressions for the second-order operators.

Now substitute these expressions into the original PDE, and tidy up the result. You should find that most of the terms cancel.