Find the eigen values and eigen functions of the integral equation : $$\phi(x)-\lambda \int_{-1}^1\cos[\pi(x-t)]\phi(t)\,dt=f(x)$$
Consider the homogeneous integral equation \begin{align*} \phi(x)&=\lambda \int_{-1}^1\cos[\pi(x-t)]\phi(t)\,dt\\ &=A\lambda \cos(\pi x)+B\lambda \sin(\pi x) \end{align*} where, $A=\int_{-1}^1\phi(t) \cos(\pi t)\,dt$ and $B=\int_{-1}^1\phi(t) \sin(\pi t)\,dt$.
Solving we get, $(1-\lambda)A=0$ and $(1-\lambda)B=0$. So $\lambda =1,1$ are the eigen values.
But I am unable to find the eigen function corresponding to the eigen value $1$.
Any help plase?