Solving ux + (x/y)uy = 0 Using Characteristics

[peripatein]

Hi,

Homework Statement

I have solved u_x + (x/y)u_y = 0 using characteristics, to obtain
u(x,y)=C (for y=+-x) and f(x²-y²)

Homework Equations

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.

 

[pasmith]

Hi,

Homework Statement

I have solved u_x + (x/y)u_y = 0 using characteristics, to obtain
u(x,y)=C (for y=+-x) and f(x²-y²)

Homework Equations

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.

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 $\epsilon \neq 0$.

 

[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 :).