An integral formula for the inhomogeneous Jordan--von Neumann equation

We study the inhomogeneous form of the Jordan–von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^2$ solution is equivalent to $g$ being itself of class $C^2$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g$ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^k$, smooth, and polynomial classes.