PDE i.v.p. using method of characteristics

[sunrah]

Homework Statement

solve x²u_x + y²u_y = 0 for u(2,y) = y

Homework Equations

The Attempt at a Solution

with a = x² and b = y²

y' = b/a = (y/x)² this can be solved for y by separation of variables:

$y = \frac{x}{1-xC}$
and
$C = \frac{y-x}{xy}$

now

$u(x,y) = f(C) = f(\frac{y-x}{xy})$

applying initial value conditions

$u(2,y) = f(\frac{y-2}{2y})$

this is where my understanding runs out. how do i determine f according to the initial value? i have looked at several books but they just assume this step is obvious

 

[sunrah]

have solved it now by taking a different approach and by converting to a system of ODEs and doing a coordinate transform from x(t) -> x(t,s) but I would still like to know how to solve it according to my original question, thanks