TITLE: A question on the method of characteristics
QUESTION [5 upvotes]: I recently came across the method of characteristics (before that I had only used separation of variables) and decided to read up on it. I tried reading the wikipedia article on it, but I'm still not clear how to get the solution! They all only seem to give the characteristics... I thought I'd start with a very simple form of their Eq.(1) as:
$$\frac{\partial u}{\partial x}+a(x,y,u) \frac{\partial u}{\partial y}=0$$
I'm not clear as to how to solve this, as it starts with "suppose a solution $u$ is known" (which I don't know!), and with $z=u(x,y)$, I end up with 
$$\begin{align}
dx/dt&=1\\
dy/dt&=a(x,y,z)\\
dz/dt&=0
\end{align}
$$
How do I proceed from here? How do I get $u(x,y)$? I know I need a boundary condition for a D.E., but I don't know what to choose... is $u(0,y)=u_0$ a sufficient BC?

REPLY [2 votes]: You can still follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$
$\dfrac{du}{dt}=0$ , letting $u(0)=u_0$ , we have $u=u_0$
The remaining is $\dfrac{dy}{dt}=a(x,y,u)=a(t,y,u_0)$ with letting $y(0)=f(u_0)$