Continuous solutions of $f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy$ 
Possible Duplicate:
Eigenvalues of an operator 


Find all the functions $f \in C([0,\frac{\pi}{2}])$ which are solutions of 
  $$
f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad \lambda \in \mathbb R.
$$

I just tried some algebraic manipulations and calculations, but I do not know ho to face this problem. Any help or hint is welcomed. Thanks a lot.

A different solution from the one by  nbubis is differentiation under the integral sign. Hence we get,
\begin{equation}
\frac{df(x)}{dx} = - \lambda \int_0^{\pi/2} \sin({x-y)}f(y)\,dy, \qquad \lambda \in \mathbb R.
\end{equation}
and then differentiating again we get
\begin{equation}
\frac{d^{2}f(x)}{dx^{2}} = - \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad \lambda \in \mathbb R.
\end{equation}
which is basically
\begin{equation}
\frac{d^{2}f(x)}{dx^{2}} = - f(x), \qquad \lambda \in \mathbb R.
\end{equation}
Hence, 
\begin{equation}
f(x) = a\cos(x) + b \sin(x)
\end{equation}
now you can put this in the integral and you will get appropriate values of $a,b$ 
