Find a solution of the equation $$ -\frac{d^2u}{dx^2} - u = -1$$ in the interval $|x| < 1$ which does not obey the maximum principle.
The maximum principle: If $u \in C^2(\Omega)$ satisfies $\Delta u \geq 0$ in $\Omega$, then either $u$ is a constant , or $u(\xi) < \sup \limits_{x \in \Omega} u (x)$ for all $\xi \in \Omega$
So, I have computed the general solution of the above equation and got the general solution as $u(x) = c_1 cos(x) + c_2sin(x) +1$ but I am struggling to find such solution that does not obey the maximum principle.
Obviously, $u(x)$ is not a constant but how should I find such solution such that the second condition $u(\xi) < \sup \limits_{x \in \Omega} u (x)$ for all $\xi \in \Omega$ does not hold.