# Second order PDE technique

Im working through some problems in Strauss book on PDE. I can't find a good method to solve the following:

$3u_x + u_{xy} = 0$. Where $u=u(x,y)$ and $u_x$ denotes the partial w.r.t $x$.

Now, the text provides no discussion on equations of this form. My instinct says to integrate over $x$ and solve the result as an ODE of a function of $y$. This is the second problem in the book though and it seems there should be a more direct approach. Any ideas?

-
$3D_x+D_{xy} = D_x(3+D_y)$ hence $u(x,y)=c_1+c_2e^{-3y}$... did I miss anything? – James S. Cook Sep 6 '12 at 18:13
@JamesS.Cook: yes, you did. The "constants" are not constant. – Robert Israel Sep 6 '12 at 18:18
@RobertIsrael ah, but fortunately I did not say they were constant :) – James S. Cook Sep 6 '12 at 18:22

Actually I'd go the other way. Solve $3 U_x + \dfrac{\partial}{\partial y} U_x = 0$ to get $U_x = C(x) e^{-3y}$. Then integrate with respect to $x$ (remembering that the "constant" of integration can depend on $y$).