Problem:

$\displaystyle u_{xx} + u_{yy} = 0, 0 < x < a, y > 0$

$\displaystyle u(0, y) = u(a, y) = 0$ and $\displaystyle u(x, 0) = f(x)$

$\displaystyle u$ bounded as $\displaystyle y \to \infty$

Find $\displaystyle u(x, y)$ using separation of variables for general $\displaystyle f(x)$ and write the solution in the form:

Eq. 1

$\displaystyle u(x,y) = \displaystyle\int_0^a G(x, y, s) f(s) ds \label{1}$

Find $\displaystyle u(x, y)$ in as explicit a form as you can, i.e., sum the series.

Attempt at solution:

After using the method of separation of variables I came to the solution:

$\displaystyle u(x, y) = \displaystyle\sum_{n = 1}^\infty c_n e^{-n\pi y / a}\sin{\frac{n\pi x}{a}}$

Where $\displaystyle c_n = \frac{2}{a}\displaystyle\int_0^a f(x) \sin{\frac{n\pi x}{a}} dx$

I'm just not sure how I'm supposed to combine these two pieces of information to be able to express $\displaystyle u(x, y)$ in the form outlined in Eq. 1.

Any assistance would be greatly appreciated!