Math 6420 • PDE

Consider the integral equation
\[
  \phi(x)-\lambda\int_0^\pi\sin(x+t)\phi(t)\,dt
  =
  \cos(x)+\sin(x),
  \qquad 0\le x\le \pi.
\]
\begin{enumerate}
\item[(a)] Find the unique solution when \(\lambda\ne\pm 2/\pi\).

\item[(b)] Use the Fredholm alternative theorem to show that there is no solution when
\(\lambda=2/\pi\).

\item[(c)] Use the Fredholm alternative theorem to show that when \(\lambda=-2/\pi\) there is a
one-parameter family of solutions of the form
\[
  \phi(x)=\frac{\cos(x)+\sin(x)}{2}+\alpha(\cos(x)-\sin(x)).
\]
\end{enumerate}