I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, and I know that I should get two eigenvalues, but I can't seem to find them.
Is there a standard method to finding the lambda's other than solving the equation $Kv(x)=v(x)$?
Thanks for the help!