Fredholm Integral Equations I'm having problems obtaining the solution of
the homogeneous Fredholm Integral Equation of the
2nd kind, with separable kernel. I always get a zero
if I use the normal method I was taught for the non
homogeneous type. I have an example: $$y(x) =
\lambda \int_{-1}^1(x+z)y(z)dz$$.
Can you help me please?

If $y$ is such a solution, then
$$
          y = \left(\lambda \int_{-1}^{1}y(z)dz\right) x + \left(\lambda\int_{-1}^{1}zy(z)dz\right)
$$
forces $y=Ax+B$ for some constants $A$ and $B$. Substituting into the above
$$
     Ax+B=\left(\lambda\int_{-1}^{1}(Az+B)dz\right)x+\left(\lambda\int_{-1}^{1}(Az^{2}+Bz)dz\right) \\
   =2\lambda Bx+\frac{2}{3}\lambda A.
$$
So it is necessary that
$$
              A=2\lambda B,\;\;\;B=\frac{2}{3}\lambda A \\
                A=\frac{4}{3}\lambda^{2}A,\;\;\; B=\frac{4}{3}\lambda^{2}B.
$$
So, either $A=B=0$, or $\lambda = \pm\frac{\sqrt{3}}{2}$. All that remains is to try the possibilities. Refer to the second equation above.
