Find the general integral of the equation (∂^2 z)/(∂ x ∂ y)…
Find the general integral of the equation \[ \frac{\partial^2 z}{\partial x\,\partial y} = 0. \]
Explanation of Method
Integrate successively, once with respect to each variable; each integration introduces one arbitrary function of the variable that was held fixed.
Step 1 · Integrate once with respect to $y$
Treating $x$ as a parameter: \[ \frac{\partial z}{\partial x} = \int 0\,dy = f(x), \] where $f(x)$ is an arbitrary differentiable function of $x$ alone.
Step 2 · Integrate once more with respect to $x$
\[ z = \int f(x)\,dx + \varphi(y) = F(x) + \varphi(y), \] where $F(x) = \int f(x)\,dx$ is again an arbitrary differentiable function of $x$, and $\varphi(y)$ is an arbitrary differentiable function of $y$.
Verification
\[ \frac{\partial z}{\partial y} = \varphi'(y), \qquad \frac{\partial^2 z}{\partial x\,\partial y} = 0. \checkmark \]
Result
The general integral is: