Math 6420 • PDE

Consider the integral equation
\[
  u(x)=f(x)+\frac{1}{2}\int_0^1 xtu(t)\,dt,\qquad\text{for }0\le x\le 1,
\]
where \(f(x)\) is a prescribed function.
\begin{enumerate}
\item[(a)] Show that the equation has at most one solution.
\item[(b)] Derive a formula for the solution that is valid for any continuous function
\(f(x)\).
\item[(c)] Calculate the solution for the special case \(f(x)=5x/6\).
\end{enumerate}
\end{enumerate}