What kind of pde is and how can I solve it? I could not find a category of PDE where I can classify this equation and I do not know how to begin to solve it.
$\Delta u - u = 1$ in $\Omega = [0,\pi]\times [0,\pi]$
with the boundary conditions:
$u = 0$ on $[0,\pi]\times\{0\}\cup[0,\pi]\times\{\pi\}$ and
$\frac{\partial u}{\partial \eta} = 0$ on $\{0\}\times[0,\pi]\cup\{\pi\}\times[0,\pi]$
I tried with separation of variables, but without results.

The conditions given result in a function $u(x,y)$ that is constant in $x$. That is, $u(x,y)=U(y)$. And,
$$
             U''(y)-U(y)=1,\;\;\; U(0)=0,\; U(\pi)=0.
$$
That can be solved with ODE methods and the annihilator method
$$
              D(D-1)(D+1)U=0.
$$
The general solution is
$$
              U(y)=-1+Be^{y}+Ce^{-y}
$$
where the constants $B,C$ are uniquely determined by the endpoint conditions.
$$              U(0)=-1+B+C=0 \\
              U(\pi)=-1+Be^{\pi}+Ce^{-\pi}=0
$$
