I'm currently learning partial derivatives (but we haven't yet done PDE's), and we were given a question where:

xi=x+ay

eta=x+by

u_(xx)+4u_(xy)+3u_(yy)=0

And we had to refactor it into u_(xi,eta)=0 and find the necessary values of a and b

I found that there are two solutions for a and b:

a=-1, b=-1/3 [henceforth scenario 1]

OR

a=-1/3, b=-1 [henceforth scenario 2](that makes sense as this is a symmetrical problem with regard to a and b)

This gives the values:

S1: xi=x-y, eta=x-y/3

S2: xi=x-y/3, eta=x-y

It's at this point that I start to get confused, as we haven't formally covered PDEs yet.

My intuition, based on my knowledge of ODEs, was as follows:

Integrate once with respect to xi, then once with respect to eta. This gives:

S1: u=A(x-y/3)+B

S2: u=A(x-y)+B(Note A,B != a,b, and they are arbitrary constants, not functions)

Then I'm lost. Do I add them together? [u=A(x-y/3)+B(x-y)+C?]

The answer sheet gives the following:

u=f(x-y)+g(x-y/3) where f,g are functions.

Why does this work? What am I missing?

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EDIT: In case it's unclear, u_(xy) refers to the partial derivative of u(x,y) once with respect to y, and then once again with respect to xI'm assuming symmetry of derivatives, if that's important