What is a general recipe for solving a first order PDE of the form $a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$, where we know $u = f(x,y)$ on some line $h(x,y) = 0$?
It would also be very helpful if somebody could provide an intuitive explanation to complement the recipe.
Thanks for any help!
You can also use the the Lagrange auxiliary equations to convert PDE to a nonlinear (actully depends on the given equation) system of first order ODEs. Let me say a couple of words for that. The Lagrange equations are $\frac{dx}{a(x,y,u)}=\frac{dy}{b(x,y,u)}=\frac{du}{c(x,y,u)}$. From this you have \begin{equation} \frac{dy}{dx}=\frac{b(x,y,u)}{a(x,y,u)}\qquad (1) \end{equation} \begin{equation} \frac{du}{dx}=\frac{c(x,y,u)}{a(x,y,u)}\qquad (2) \end{equation}
The system (1)-(2) is not easy to solve in general. Once solving this system you find $y=y(x,c_1)$ and $u=u(x,c_2)$ where $c_1$ and $c_2$ are any integration constants. Eliminating these $c_1$ and $c_2$ you come up with a general solution (called integral surface) given implicitly $G(x,y,u)=0$ for some continuosly diferentiable function $G$. If you have a Cauchy (or initial-value problem) $c_1$ and $c_2$ can be determined by this data.