Math 6420 • PDE

\begin{enumerate}
\item[(a)] By constructing the one-dimensional Green's function solve the inhomogeneous equation
\[
  \frac{d^2u}{dx^2}=f(x),\qquad u(0)=\alpha,\qquad u(1)=\beta.
\]

\item[(b)] Use Green's functions to solve
\[
  \frac{d^2u}{dx^2}-\alpha^2u=f(x)\quad\text{on }L^2(-\infty,\infty)
\]
with \(\alpha\) real.
\end{enumerate}