Find the general integral of the equation r = xy, r = (∂^2…
Find the general integral of the equation \[ r = xy, \qquad r = \frac{\partial^2 z}{\partial x^2}. \]
Explanation of Method
Integrate $\dfrac{\partial^2 z}{\partial x^2} = xy$ twice with respect to $x$, treating $y$ as a parameter; each integration introduces an arbitrary function of $y$.
Step 1 · First integration with respect to $x$
\[ \frac{\partial z}{\partial x} = \int xy\,dx = \frac{x^2 y}{2} + \varphi(y), \] where $\varphi(y)$ is an arbitrary differentiable function of $y$.
Step 2 · Second integration with respect to $x$
\[ z = \int\!\left(\frac{x^2 y}{2} + \varphi(y)\right)dx = \frac{x^3 y}{6} + x\,\varphi(y) + \psi(y), \] where $\psi(y)$ is a second arbitrary differentiable function of $y$.
Verification
\[ \frac{\partial z}{\partial x} = \frac{x^2 y}{2} + \varphi(y), \qquad \frac{\partial^2 z}{\partial x^2} = xy. \checkmark \]
Result
The general integral is: