By using the method of characteristics, show that the solution of
the problem
xu(x) + yu(y) - zu(z) = u;
u(1; y; z) = z
and u(x; 1; z) = zx^2

is given by u(x; y; z) = zx^2.

My question is how to I solve questions of this nature?

I know I start with

ds/dx = x

ds/dy = y

ds/dz= -z

du/ds = u

I know I have to integrate all first order equations but then bringing them together and using the boundaries is where I get lost?