Help me to find the second solution

Question

I'm trying to complete a exercise using Lagrange's Linear Equations. I got one solution but I'm unable to find a way to get the second solution.
This is the question that I'm trying to solve:
$$ \left(x+y\right)\cdot\left(p+q\right) = z - 1 $$ Here it is what I have tried so far:

Answer 1

Hint: you found that $x-y=c \implies y=x-c, x+y=2x-c$.

Your second solution will come from solving $$\frac{dx}{2x-c}=\frac{dz}{z-1}$$