Find the general integral of the equation pq = zs, p = (∂…

Find the general integral of the equation \[ pq = zs, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y},\quad s = \frac{\partial^2 z}{\partial x\,\partial y}. \]

Step 1 · Rewrite in a suggestive form

Observe that \[ \frac{\partial^2(\ln z)}{\partial x\,\partial y} = \frac{\partial}{\partial x}\!\left(\frac{q}{z}\right) = \frac{sz - qp}{z^2} = \frac{s}{z} - \frac{pq}{z^2}. \] Using $pq = zs$, $pq/z^2 = s/z$, so \[ \frac{\partial^2(\ln z)}{\partial x\,\partial y} = \frac{s}{z} - \frac{s}{z} = 0. \tag{2} \]

Step 2 · Integrate equation (2)

Equation (2) is exactly Problem 11's equation for $\ln z$, whose general integral is \[ \ln z = f(x) + g(y), \] with $f(x)$, $g(y)$ arbitrary differentiable functions. Exponentiating and writing $\Phi(x)=e^{f(x)}$, $\Psi(y)=e^{g(y)}$ (both arbitrary): \[ z = \Phi(x)\,\Psi(y). \]

Verification

$p=\Phi'(x)\Psi(y)$, $q=\Phi(x)\Psi'(y)$, $s=\Phi'(x)\Psi'(y)$, so \[ pq = \Phi(x)\Phi'(x)\,\Psi(y)\Psi'(y) = zs. \checkmark \]