Problem 120
Easy Separation of Variables

Find the complete integral of p^2-q^2 = 1

Find the complete integral of \[ p^2-q^2 = 1. \]

Explanation of Method

Set $q=a$, solve for $p$, integrate.

Step 1 · Set $q=a$ and integrate

\[ p=\sqrt{1+a^2} \implies z = x\sqrt{1+a^2}+ay+b. \]

Verification

$p^2-q^2=(1+a^2)-a^2=1$. \checkmark

Result

The complete integral is:

\[ \boxed{ z = x\sqrt{1+a^2}+ay+b } \]