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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?

2 Answers 2