When does the equation $\frac{dy}{dx} = \frac{b}{a}$ represents the slope of the projection of the characteristic curve of $au_x + bu_y = c$ onto the $x,y$ plane and why? How is equation obtained from the characteristic equations?
In general, the method of characteristics works for $a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$ (quasi-linear PDEs) and allows one to build the solution surface as a union of characteristics. What I can't understand is how we can move from the characteristic equations $$ \frac{dx}{dt} = a , \, \frac{dy}{dt}=b, \, \frac{du}{dt}=c, $$ to the equations $\frac{dy}{dx} = \frac{b}{a}, \, \frac{du}{dx} = \frac{c}{a}$ or even more generally to $$ \frac{dx}{a}=\frac{dy}{b} = \frac{du}{c}. $$ I guess these have something to do with the chain rule, but can't find any good resource that explains it thoroughly.
Hope I stated the question clearly enough. Thank you
The first point $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{b}{a}$$ The second point: $$\frac{du}{dx}=\frac{\frac{du}{dt}}{\frac{dx}{dt}}=\frac{c}{a}$$
Basically, a symmetric form of ODE system is just a convenient way to write it down. Say, we have a system of n ODEs $$\frac{dx_1}{ds} = F_1(x_1, ..., x_n),...,\frac{dx_n}{ds} = F_n(x_1, ..., x_n)$$ We can rewrite it as $$\frac{dx_1}{F_1(x_1, ..., x_n)} = ds,...,\frac{dx_n}{F_n(x_1, ..., x_n)} = ds$$ Now each expression is equal to $ds$, so we can rewrite it as a chain equality: $$\frac{dx_1}{F_1(x_1, ..., x_n)} = ... = \frac{dx_n}{F_n(x_1, ..., x_n)} = ds$$