Use separation of variables with the given Cauchy data to solve a PDE 
Consider $\Delta u=0$ in $\mathbb{R}^2,$ with the Cauchy data: 
  $$u=0, \frac{\partial u}{\partial x_2} = \frac{1}{n} \sin(n x_1),~~\textrm{on}~\{ X_2 = 0 \}.$$

My approach: WLOG let $X_1(x_1)=X(x)$ and $X_2(x_2)=Y(y).$ Now separation of variables, $u(x , y)=X(x) Y(y)$ yields:
$$\frac{X''(x)}{X(x)}=-\frac{Y''(y)}{Y(y)}=-\lambda~~\textrm{for}~\lambda >0.$$
Now solving the two ODEs yield:
$$X(x)=A \cos \sqrt{\lambda}x + B \sin \sqrt{\lambda}x~~,~~Y(y)=C \cosh\sqrt{\lambda}y + B \sinh \sqrt{\lambda}y.$$
Can someone please explain me how to use the Cauchy data to complete the given problem. Thank you for your time.

HINT :
$$u(x,y)=\left(A \cos \sqrt{\lambda}x + B \sin \sqrt{\lambda}x\right)\left(C \cosh\sqrt{\lambda}y + D \sinh \sqrt{\lambda}y\right).$$
$$\begin{cases} 
\left(\frac{\partial u}{\partial y}\right)_{y=0} =\left(A \cos \sqrt{\lambda}x + B \sin \sqrt{\lambda}x\right)D\sqrt{\lambda} 
=\frac{1}{n} \sin(n x) \quad\to\quad 
\begin{cases}
\sqrt{\lambda}=? \\
A=? \\
BD=?
\end{cases} \\
u(x,0)=\left(A \cos \sqrt{\lambda}x + B \sin \sqrt{\lambda}x\right)C=0 \quad\to\quad C=?
\end{cases}$$
