Problem 115
Find the complete integral of z = px+qy+p^2+q^2
Find the complete integral of \[ z = px+qy+p^2+q^2. \]
Explanation of Method
This has the Clairaut form $z=px+qy+f(p,q)$; the complete integral is obtained immediately by replacing $p,q$ with arbitrary constants $a,b$.
Step 1 · Apply the Clairaut substitution
Set $p=a$, $q=b$: \[ z = ax+by+a^2+b^2. \]
Verification
$p=a$, $q=b$, so $px+qy+p^2+q^2=ax+by+a^2+b^2=z$. \checkmark
Result
The complete integral is:
\[ \boxed{ z = ax+by+a^2+b^2 } \]