Problem 113
Find the complete integral of p = 3q^3, p=(∂ z)/(∂ x), q=(∂…
Find the complete integral of \[ p = 3q^3, \qquad p=\frac{\partial z}{\partial x},\ q=\frac{\partial z}{\partial y}. \]
Explanation of Method
For an equation $F(p,q)=0$ containing no $x,y,z$ explicitly, set $q=a$ (arbitrary constant), solve for $p$, and integrate $dz=p\,dx+q\,dy$.
Step 1 · Set $q=a$ and solve for $p$
\[ p = 3a^3. \]
Step 2 · Integrate
\[ z = \int 3a^3\,dx+\int a\,dy = 3a^3x+ay+b. \]
Verification
$p=3a^3$, $q=a$, so $3q^3=3a^3=p$. \checkmark
Result
The complete integral is:
\[ \boxed{ z = 3a^3x+ay+b } \]