Solving a Second Order PDE I'm trying to solve the equation $u_t = \alpha^2 U_{yy}$  given $u(y,t)$ bounded $y \rightarrow\infty$ and $u(0,t) = U_o e^{iw_ot}$. Initial is $u(y,0) = 0$. I have gotten both separations as $Y'' - \lambda Y=0$ and $T' = \alpha^2T$, but from here I get confused what to do, I never learnded PDE and am trying to solve a model. Thanks.

I would like to point out that if $u(y,0)=0$, the solution is $u(y,t)=0$. This is a classical Possion problem which is derive from heat diffusion. So this equation is also called heat equation. I think a simple way to solve this equation is to use Fourier Transformation.
Let $\hat{u}(x,t)=F[u(y,t)]$ denote the corresponding spatial Fourier transformation of $u$. Assume the initial condition is not zero, say $u(y,0)=\phi(y)$. Then the equation can be written by:
\begin{equation}
\hat{u}_t(x,t)+\alpha^2y^2\hat{u}(x,t)=0\\
\hat{u}(x,0)=\hat{\phi}(x)
\end{equation}
This is a classical 1st order ODE initial problem. The solution is 
\begin{equation}
\hat{u}(x,t)=\hat{\phi}(x)e^{-a^2x^2t}
\end{equation}
Then take the inverse Fourier transformation (denoted by $F^{-1}[u(x,t)]$), we obtain the solution of the original PDE:
\begin{equation}
u(y,t)=F^{-1}[\hat{\phi}(x)e^{-a^2x^2t}]=\phi(y)*F^{-1}[e^{-a^2x^2t}]\\
=\frac{1}{2\alpha\sqrt{\pi t}}\int_{-\infty}^{+\infty}\phi(\xi)e^{-\frac{(y-\xi)^2}{4\alpha^2t}}d\xi
\end{equation}
But this mehtod is limited because it assume the spatial boundary of region of $u(y,t)$ is zero then the $u$ is an element of Schwartz space which made Fourier transformation is valid. If the spatial boundary is not zero, you can use the technique suggested by Heat Equation. 
