Why does the solution of this Laplace Equation need to be of the form $\sum_{n\geq 1} \sin{nx} [A_n \sinh{(ny)} + B_n \sinh{(n(y- \pi))}]$?

Question

Let $u$ solve the Laplace equation with Dirichlet boundary conditions:

$$\Delta u = 0 \\u(x,0)=\sin x \\u(x,\pi)=\sin x + 1/2 \sin(2x) \\ u(0,y) = u(\pi,y )=0$$ with $0 \leq x,y \leq \pi$

We want to find the explicit formula for $u$. We look for a solution of the form $$u(x,y)= \sum_{n\geq 1} \sin{nx} [A_n \sinh{(ny)} + B_n \sinh{(n(y- \pi))}]$$

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May I ask you why the solution needs to be like $\sum_{n\geq 1} \sin{nx} [A_n \sinh{(ny)} + B_n \sinh{(n(y- \pi))}]$ ?