# PDE question.

1. Dec 31, 2013

### peripatein

Hi,
1. The problem statement, all variables and given/known data
I have solved ux + (x/y)uy = 0 using characteristics, to obtain
u(x,y)=C (for y=+-x) and f(x2-y2)

2. Relevant equations

3. The attempt at a solution
I was then given two boundary conditions:
(a) u(x=0,y)=cos(y), which I used to obtain u(x,y) = cos(√(y2-x2))
(b) u(x=y,y)=y
I am not quite sure how to approach this latter and would appreciate some advice.

2. Dec 31, 2013

### pasmith

The problem is not well-posed: you've shown that $y = x$ is a characteristic, so $u$ must be constant on $y = x$ and the value of $u$ on $y = x$ tells you nothing about the value of $u$ on $x^2 - y^2 = \epsilon$ for any $\epsilon \neq 0$.

3. Dec 31, 2013

### peripatein

Pardon me, I did mean to add that. Indeed, if C=0 then, for y=+-x, u(x,y) = C. Otherwise (C different than zero), u(x,y)=x^2-y^2.
I'd still appreciate your help with my problem :).