PDE Characteristic Equations - Remembering Them? Is there a quick smart intuitive way to see that $$\frac{dx}{F_p} = \frac{dy}{F_q} = \frac{du}{pF_p+qF_q} = - \frac{dp}{F_x+pF_u} = - \frac{dp}{F_y+qF_u}$$ are the characteristic equations for a non-linear pde $F(x,y,u,u_x,u_y)= 0$? Doesn't even matter how illogical or inapplicable so long as it gives the right result. If I forget them, it'd just take absolutely _ages_ to re-derive them in an exam.

Maybe it's not that intuitive or even easy to remember, but you may want to have a look at the most general case of the _Lagrange-Charpit equations_ :

$$ \frac{\mathrm{d}x_i}{F_{p_i}}=-\frac{\mathrm{d}p_i}{F_{x_i}+F_u \, p_i}=\frac{\mathrm{d}u}{\sum_i \, p_iF_{p_i}}, $$ which are the characteristic equations for the non-linear pde:

$$ F(x_1,x_2, \ldots, p_1,p_2,\ldots, u) = 0, $$

where $p_i = u_{x_i}$ and $x_i$ are the set of indepent variables, whilst $u$ is the unknown.

Cheers!

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