I've been reading about the method of characteristics and came across the theorem:
The general solution of a first-order, quasi-linear partial differential equation
$$
a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)\tag{2.5.1}
$$
is
$$
f(\phi,\psi)=0,\tag{2.5.2}
$$
where $f$ is an arbitrary function of $\phi(x,y,u)$ and $\psi(x,y,u)$, and $\phi=\text{constant}=c_1$ and $\psi=\text{constant}=c_2$ are solution curves of the characteristic equations
$$
\frac{dx}{a}=\frac{dy}{b}=\frac{du}{c}.\tag{2.5.3}
$$
The solution curves defined by $\phi(x,y,u)=c_1$ and $\psi(x,y,u)=c_2$ are called the families of characteristic curves of equation $(2.5.1)$.
I've gone through the proof and it seems straightforward. But I'm unable to visualize what the characteristic curves $\phi=c_1$ and $\psi=c_2$ represent in the $(x,y,u)$ space(can they be any curves on the solution surface or they follow a certain property?) and why $(2.5.2)$ intuitively should give me the general solution.
Best Answer
Consider fixed $c_1$ and $c_2$, then these two equations determine a curve. Now, let $c_1$ run free but remaining $c_2$ fixed, obviously we have a surface and it is a solution of the PDE. This gives us a tip about how the particular solutions emerge (we got one!). We can do better, we can move $c_1$ and simultaneously move $c_2$, we draw a surface that is solution too. But the simultaneous movement of $c_1$ and $c_2$ is what we call "function" and because this function is not still determined, we say that it is arbitrary. So $c_2=f(c_1)$ or $\psi=f(\phi)$
A particular solution determines $f$. Consider this example: we know the value of $u$ along the line $y=0$, for each $x$ we know $u$, i. e. $u$ is a perfectly known function $g$ of $x$ $u=g(x)$ along $y=0$ (you can imagine $x^2$ or $e^x$). Then each of the curves have to satisfy that requirement, so is, $\phi(x,0,g(x))=c_1$ and $\psi(x,0.g(x))=c_2$. Now, $x$ can be eliminated determining the needed functional relation between $c_1$ and $c_2$