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Show that the complete integral of pde $F(u,p,q)=0$ ($p=u_{x}$ and $q=u_{y}$) is $$ f(x,y,u,a,b) = x + ay + b - \int\frac{du}{g(u,a)}, $$ where the function $p=g(u,a)$ is computed from the differential equation. I have started with writing Charpit system, namely: $$ \begin{cases} \frac{dX}{ds} = F_{p} \\ \frac{dY}{ds} = F_{q} \\ \frac{dP}{ds} = -F_{u}P \\ \frac{dQ}{ds} = -F_{u}Q \\ \frac{dU}{ds} = PF_{p} + QF_{q} \end{cases} $$ where $X=X(s), Y=Y(s)$ is some curve in the $x-y$ plane which is parameterized with the $s$ and $U(s)=u(X(s),Y(s))$, $P(s)=u_{x}(X(s),Y(s))$, $Q(s)=u_{y}(X(s),Y(s))$. I don't know what to do next - how can I find the solution without more knowledge about the function $F$? Thanks in advance for any help!

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Some information is missing; without knowing the structure of $F(u,p,q)$ you can't tell the structure of of $f(x,y,u,a,b)$ as presented. There could have been some prior information before the part of $F$ that you posted.

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  • $\begingroup$ perhaps more suited as a comment? $\endgroup$ Jun 13 '19 at 5:21
  • $\begingroup$ Was not allowed to post a comment!! $\endgroup$ Jun 13 '19 at 7:42

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