Problem 117
Find the complete integral of z = px+qy+p^3q^3
Find the complete integral of \[ z = px+qy+p^3q^3. \]
Explanation of Method
Clairaut form $z=px+qy+f(p,q)$; substitute $p=a$, $q=b$.
Step 1 · Apply the Clairaut substitution
\[ z = ax+by+a^3b^3. \]
Verification
$p=a$, $q=b$, so $px+qy+p^3q^3=ax+by+a^3b^3=z$. \checkmark
Result
The complete integral is:
\[ \boxed{ z = ax+by+a^3b^3 } \]