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