I have the following integral equation with symmetric kernel $$ x(t)=\sin(\pi t)+\pi \cos (\pi t) +\lambda \int_{0}^{1} k(t,s)x(s)\,ds $$ where $k(x,t)$ is a symmetric kernel given by $$k(t,s)= \begin{cases} (t+1)s, & 0 < t <s \\ (s+1)t, & s < t <1 \end{cases} $$ I want to find its eigenvalues and its eigenfunctions.
The first thing I do is to consider the homogeneous problem, i.e. the associated homogeneous integral equation $$ x(t)=\lambda\int_{0}^{1}k(t,s)x(s)ds. $$ By taking the derivative with respec to $t$ twice in this equation I obtain the following second order ordinary differential equation $$ x''(t)=\lambda x(t) $$ which with the boundary conditions $x(0)=x'(0)$ and $x(1)=x'(1)$ becomes the following Robin boundary problem $$ \begin{cases} x''(t)=\lambda x(t)\\ x(0)=x'(0)\\ x(1)=x'(1) \end{cases} $$ I quickly solved this problem, but now I am not sure on how to procede in order to solve the former eigenvalue/eigenfunction problem. I know that is a Sturm-Liouville problem, but I'm somewhat confused on how to deal with this kind of boundary conditions.
I will be very grateful for any hint or help.