Solve the integral equation
$$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$
where $f$ is continuous using the method of finding the resolvent kernel and Newmann series.
Here it is what I did:
$ K_1 (t,s) \equiv K(t,s) =t-s$
$ K_2 (t,s) = \int_{s}^{t} K(t, \xi) K_1 (\xi ,s) d \xi= \frac{1}{2} (t+s)^2(t-s)-ts(t-s) +\frac{1}{3} (s^3 -t^3) $
From here and on the calculations are too difficult.
Is there any trick?
Any help?
Thank's in advance!
P.S Is there another way to solve it (without using this method) ?
edit: I didn't made any proccess. Some help?
No need to expand the integrand. Linear change of variables mapping $[s,t]$ to $[0,1]$ reduces integrals for $K_n$ to beta-function. It will be easy to calculate several first ones, guess the formula for $K_n$ and prove it by induction.
Edit
Making change of variables $\xi=s+y(t-s)$ we have $$ \int_s^t(t-\xi)(\xi-s)\,d\xi= (t-s)^3\int_0^1(1-y)y\,dy= (t-s)^3B(2,2). $$