I need to solve the following by using the method of characteristics $u\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=1~,~u|_{x=y}=\frac{x}{2}$
I have the following characteric equations:
$\frac{dx}{ds}=u~;\frac{dy}{ds}=1~;\frac{du}{ds}=1$
from the above I get
$ x=us+x_{0} $ $ y=s +y_{0} $ $ u=s+u_{0} $
I am now thinking I should go with the standard conditions $y_0=0$ and $u(x,0)=f(x_0)$
this now gives me: $ x=uy+x_o $ $ y=s $ $ u=y+f(x_0) $
Im confused because of the $u$ term in my equation for $x$
Can anyone please help.
Thanks a mil