Problem 122
Find the complete integral of p^2+q^2 = 9
Find the complete integral of \[ p^2+q^2 = 9. \]
Explanation of Method
Set $q=a$, solve for $p$, integrate.
Step 1 · Set $q=a$ and integrate
\[ p=\sqrt{9-a^2}\ (|a|<3) \implies z = x\sqrt{9-a^2}+ay+b. \]
Verification
$p^2+q^2=(9-a^2)+a^2=9$. \checkmark
Result
The complete integral is:
\[ \boxed{ z = x\sqrt{9-a^2}+ay+b,\quad |a|<3 } \]