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PDE : Complete solution to a non-linear equation.

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The PDE is: xp + yq + p + q -pq = u

which gives: p(x+1) + q(y+1) - pq - u = 0

Now, I need to find a complete solution. I have set up my characteristic system to be:

dx /x+1-q = dy /y+1-p = du /u-pq = dp /p(pq-u-1) = dq /q(pq-u-1)

Help! I cannot solve any of these integrals. I need p = P(x,y,u,a) and q = Q(x,y,u,a) so that I can solve du = Pdx + Qdy to obtain g(x,y,u,a) = b which would then give me u=U(x,y,a,b) a complete integral! I really need help in solving these integrals. Thank you for your help.

PLease note that I will use * to indicate a partial derivative. Thus, u*x denotes the partial derivative of u with respect to x. In addition, I will abbreviate u*x with p and u*y with q. Thus, u*x=p and u*y=q. Also, the symbol / means division. Here is the problem:

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