Find the general integral of the equation s = x^2 + y^2, s =…
Find the general integral of the equation \[ s = x^2 + y^2, \qquad s = \frac{\partial^2 z}{\partial x\,\partial y}. \]
Explanation of Method
Integrate $\dfrac{\partial^2 z}{\partial x\,\partial y} = x^2 + y^2$ first with respect to $y$, then with respect to $x$, introducing one arbitrary function at each step.
Step 1 · First integration with respect to $y$
\[ \frac{\partial z}{\partial x} = \int\!\left(x^2 + y^2\right)dy = x^2 y + \frac{y^3}{3} + \varphi(x), \] where $\varphi(x)$ is an arbitrary differentiable function of $x$.
Step 2 · Second integration with respect to $x$
\[ z = \int\!\left(x^2 y + \frac{y^3}{3} + \varphi(x)\right)dx = \frac{x^3 y}{3} + \frac{x y^3}{3} + \Phi(x) + \psi(y), \] where $\Phi(x) = \int \varphi(x)\,dx$ is again an arbitrary differentiable function of $x$, and $\psi(y)$ is an arbitrary differentiable function of $y$ arising from the second integration.
Verification
\[ \frac{\partial z}{\partial x} = x^2 y + \frac{y^3}{3} + \Phi'(x), \qquad \frac{\partial^2 z}{\partial x\,\partial y} = x^2 + y^2. \checkmark \]
Result
The general integral is: