Solving PDE using Method of Characteristics 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

Your solution to characteristic equations is incorrect, which you can easily check by plugging your current solution back in. The source of the problem is that $u(s)$ is not constant, thus $x^\prime(s) = u(s)$ is not solved by $x(s) = u(s) s + x_0$.
It may help to note that $x^{\prime\prime}(s) = u^\prime(s) = 1$. Now that you are back into ODE with constant coefficients, finding solutions should be easy.
