Find the complete integral of pq = 2xy, p=(∂ z)/(∂ x), q=(∂…
Find the complete integral of \[ pq = 2xy, \qquad p=\frac{\partial z}{\partial x},\ q=\frac{\partial z}{\partial y}. \]
Explanation of Method
Separate the equation into a function of $x,p$ alone equal to a function of $y,q$ alone, set both equal to an arbitrary constant $a$, solve for $p$ and $q$, and integrate $dz=p\,dx+q\,dy$.
Step 1 · Separate variables
Rewrite $pq=2xy$ as $\dfrac{p}{x}=\dfrac{2y}{q}$ (dividing both sides by $x$), and set each side equal to a constant $a\neq0$: \[ p = ax, \qquad q = \frac{2y}{a}. \] Consistency check: $pq=ax\cdot\dfrac{2y}{a}=2xy$. \checkmark
Step 2 · Integrate $dz=p\,dx+q\,dy$
\[ z = \int ax\,dx+\int\frac{2y}{a}\,dy = \frac{a}{2}x^2+\frac{1}{a}y^2+b, \] where $b$ is a second arbitrary constant.
Verification
\[ p=\frac{\partial z}{\partial x}=ax,\qquad q=\frac{\partial z}{\partial y}=\frac{2y}{a}, \qquad pq = ax\cdot\frac{2y}{a} = 2xy. \checkmark \]
Result
The complete integral is: