Math 6420 • PDE

Consider the boundary value problem \(u''=f(x)\), \(u'(0)=0\),
\(u(1)=0\) where \(f(x)\) is a prescribed function.
\begin{enumerate}
\item[(a)] Show that the problem has at most one solution.
\item[(b)] Show that the solution can be written as
\[
  u(x)=\int_0^1 k(x,y)f(y)\,dy
\]
for all continuous functions \(f(x)\) where
\[
k(x,y)=
\begin{cases}
x-1, & \text{for }0\le y\le x\le 1,\\
y-1, & \text{for }0\le x\le y\le 1.
\end{cases}
\]
\end{enumerate}