Problem 123
Find the complete integral of pq+p+q = 0
Find the complete integral of \[ pq+p+q = 0. \]
Explanation of Method
Set $q=a$, solve for $p$, integrate.
Step 1 · Set $q=a$ and solve for $p$
\[ p(a+1)=-a \implies p = -\frac{a}{a+1}\quad(a\neq-1). \]
Step 2 · Integrate
\[ z = -\frac{a}{a+1}\,x+ay+b. \]
Verification
\[ pq+p+q = -\frac{a^2}{a+1}-\frac{a}{a+1}+a = -\frac{a(a+1)}{a+1}+a = -a+a=0. \checkmark \]
Result
The complete integral is:
\[ \boxed{ z = -\frac{a}{a+1}\,x+ay+b } \]