Question 2.40: Find a one-parameter family of the first order PDE pq = x+y+......

One easily finds that

p-q+x-y=2 a \quad \text { and } \quad \frac{p+1}{q+1}=b

are two first integrals which are not in involution. Solving in p and q gives

p=\frac{1}{b-1}(b(y-x)+2 a b-b+1), \quad q=\frac{1}{b-1}(y-x+2 a-b+1).

Since necessarily \frac{\partial p}{\partial y}=\frac{\partial q}{\partial x}, It follows that b = -1 and

2p = y – x + 2a – 2, 2q = x- y – 2a – 2.

Hence the sought after one-parameter family is

\begin{aligned}z & =p q-x-y \\& =-\frac{(x-y)^2}{4}+a(x-y)-(x+y)+1-a^2\end{aligned}