Example 9.11. Let $\Omega$ be the unit square. Let $$f(x,y) = e^{x+y}[4\cos(4x)+4\cos(4y)-16\sin(4x)-16\sin(4y)]$$ with Dirichlet boundary condition defined by $g:=u|_{\partial\Omega}$ with $$u(x,y) := -\sin(4x)-\sin(4y)$$ Discussion: The exact solution of this boundary-value problem is given by $u$.
[Source: $\small\textit{Unknown - I do not have the full text} ]$
How?
According to the text, the Poisson equation is $$-(u_{xx}+u_{yy})=f$$ I even tried working backwards - with no success, understandably.
Edit: Maybe it should be this instead?: $$f(x,y) = [4\cos(4x)+4\cos(4y)-16\sin(4x)-16\sin(4y)]$$