# PDE i.v.p. using method of characteristics

## Homework Statement

solve x2ux + y2uy = 0 for u(2,y) = y

## 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

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