I am stuck on the following problem that says:
Find the complete integral of $z=px + qy +pq$, where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$.
My Attempt:
The given equation is : $f(x,y,z,p,q)=px+qy+pq-z$. So, Charpit's auxiliary equations are given by: $$ds={dp \over 0}={dq \over 0}= {dz \over z+pq}={dx \over x+q}={dy \over y+p}$$ Now, from $$ds={dp \over 0}, ds={dq \over 0} \implies p=C, q=D $$ being arbitray constants. Now, I have to use $$dz=pdx+qdy=Cdx+Ddy$$ we get $$z(x,y)=Cx+Dy+E$$ Now, I am stuck, because we need to have some relationship between $C$ and $D$ to have only two independent constants
I think we should use Charpit's equations to get the relation.
Can someone help me out? Thanks in advance for your time.
Inserting the constants $p=C$ and $q=D$ into the original equation you already get $$ z=Cx+Dy+CD $$ so that $E=CD$.
Additionally, $(y+C)=A(x+D)$ so that also $z+CD=(C+DA)(x+D)$.