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.

  • $\begingroup$ The book Physics for mathematicians by Michael Spivak has a nice derivation of the characteristic equations to a first-order PDE. HTH $\endgroup$ Jun 15 '15 at 15:59

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.



  • $\begingroup$ I picked the two-variable case for simplicity but if you have a nice intuitive way of remembering the form of those equations I'm all for it. $\endgroup$
    – bolbteppa
    May 9 '14 at 10:43
  • $\begingroup$ Hi there again, @bolbteppa. I'm afraid I have no other more intuitive way to recall these equations. The ones above are those which I usually remember when I need them for a particular 2D or 3D case. $\endgroup$
    – Dmoreno
    May 9 '14 at 10:53

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