Problem 121
Find the complete integral of pq+qx = y
Find the complete integral of \[ pq+qx = y. \]
Explanation of Method
Factor $q$ out to isolate $p+x$, set that combination equal to an arbitrary constant $a$, solve for $p$ and $q$, and integrate.
Step 1 · Factor and set $p+x=a$
\[ q(p+x)=y \implies q=\frac{y}{p+x}. \] Set $p+x=a$ (arbitrary constant), so $p=a-x$ and $q=y/a$.
Step 2 · Integrate
\[ z = \int(a-x)\,dx+\int\frac{y}{a}\,dy = ax-\frac{x^2}{2}+\frac{y^2}{2a}+b. \]
Verification
$p=a-x$, $q=y/a$: \[ pq+qx = \frac{y}{a}(a-x)+\frac{xy}{a} = \frac{ay}{a} = y. \checkmark \]
Result
The complete integral is:
\[ \boxed{ z = ax-\frac{x^2}{2}+\frac{y^2}{2a}+b } \]