Practice Problem #3

Here's number 3 from the practice problems. Could someone walk me through it?

Recall that the solution to Laplace's equation $\Delta$u = 0 on the unit square with boundary conditions $$u(x,0)=f(x), u(x,1)=g(x)\mbox{, and }u(0,y)=u(1,y)=0$$ is $$u(x,y)=\sum\limits_{n=1}^\infty(c_ne^{-n\pi*y}+d_ne^{n\pi*y})sin(n\pi*x),$$ where $c_n+d_n$ are the Fourier sine coefficients of f and $c_ne^{-n\pi}+d_ne^{n\pi}$ are the Fourier sine coefficients of g. Use this to solve Laplace's equation on the unit square subject to the boundary conditions $u(x,0)=1$ and $u(1,y)=u(x,1)=u(0,y)=0$. Note: It is sufficient to specify $c_n$ and $d_n$ as the unique solutions to a pair of equations.

Here is my understanding of the problem. Can someone else confirm this? It's a lot less complicated if you don't try to solve for $c_n$ and $d_n$ explicitly. Given that $c_n$ and $d_n$ are the Fourier sine coefficients of $f$, we can set up an equation to solve for the coefficients as we have done before.
Instead of $a_n=\frac{2}{\ell}\int_0^\ell f(x) \sin(n\pi x)\ dx$, the equation becomes $for $\ell=1$$
$$c_n+d_n=2\int_0^1 f(x)\sin(n\pi x)\ dx$$
In this case, we have two coefficients that we are solving for, instead of one. Since $f(x)=1$,
$$c_n+d_n=2\int_0^1 \sin(n\pi x)\ dx$$
and
$$c_n+d_n=-\frac{2}{n\pi}\cos(n\pi x)|_0^1=-\frac{2}{n\pi}\left(\cos(n\pi)-1\right)$$ .
Thus,
$$c_n+d_n=-\frac{2}{n\pi}\left((-1)^n-1\right)$$

Using a similar process, set $c_n e^{-n\pi}+d_n e^{n\pi}$ equal to the Fourier computation for $g$, so
$$c_n e^{-n\pi}+d_n e^{n\pi}=2\int_0^1 g(x)\sin(n\pi x)\ dx.$$
However, $g(x)=0$, so
$$c_n e^{-n\pi}+d_n e^{n\pi}=0.$$

Without solving for $c_n$ and $d_n$ directly, the unique solutions $c_n$ and $d_n$ can be found in the following system of equations:
$$c_n+d_n=-\frac{2}{n\pi}\left((-1)^n-1\right)$$
$$c_n e^{-n\pi}+d_n e^{n\pi}=0$$

Thanks so much, Massey! That makes perfect sense.

For everyone following along, this problem looks like an abbreviated version of another problem here on Discourse (http://www.marksmath2.org/t/relatively-easy-laplace-equation-on-a-square/96/), where @shill2 has solved for $c_n$ and $d_n$.