Problem 118
Easy Separation of Variables

Find the complete integral of pq = 9z^2

Find the complete integral of \[ pq = 9z^2. \]

Explanation of Method

Try $z=X(x)Y(y)$, giving $\dfrac{X'}{X}\cdot\dfrac{Y'}{Y}=9$; split the constant $9$ between the two logarithmic-derivative equations and integrate.

Step 1 · Substitute $z=XY$

With $p=X'Y$, $q=XY'$: \[ X'Y\cdot XY' = 9X^2Y^2 \implies \frac{X'}{X}\cdot\frac{Y'}{Y}=9. \] Set $X'/X=a$, $Y'/Y=9/a$ for arbitrary $a\neq0$.

Step 2 · Integrate

\[ X = C_1e^{ax}, \qquad Y = C_2e^{9y/a}. \]

Step 3 · Write the complete integral

With $C_1C_2=e^b$: \[ z = e^{ax+9y/a+b}. \]

Verification

$p=az$, $q=9z/a$, so $pq=az\cdot9z/a=9z^2$. \checkmark

Result

The complete integral is:

\[ \boxed{ z = e^{\,ax+9y/a+b} } \]