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 \(\lambda\ne\pm2/\pi\). Why is a unique solution
guaranteed to exist?

\item[(b)] Show that there is no solution when \(\lambda=2/\pi\).

\item[(c)] Show that when \(\lambda=-2/\pi\) there is a one-parameter family of solutions of
the form
\[
  \phi(x)=f_p(x)+\alpha f_h(x).
\]
Determine \(f_h(x)\).
\end{enumerate}