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 :).