# Volterra integral equation of second type

Solve the Volterra integral equation of second kind

$$y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds$$

I know two methods for such integral equations:

1. Picard's method

2. The mthod of finding the resolvent kernel and the the Neumann series

I tried using both of these methods but I couldn't solve it.

Which of the these methods is better to use to do the least calculations?

Thank's in advance!

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## 1 Answer

Multiply both sides of the equation with $(2t+1)^2$ and differentiate with respect to $t$: $$\frac{\mathrm{d}}{\mathrm{d} t} \left( (2t+1)^2 (y(t)-1) \right) = 2 (2t+1) y(t)$$ Now you reduced the problem to 1-st order ODE: $$\left(2t+1 \right) \left( (2t+1) y^\prime(t) + 2 y(t) - 4 \right) = 0, \qquad y(0) = 1$$

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Thank you for your time! Finally I didn't need to use the methods of solving Volterra integral equations of second type. –  passenger May 2 '12 at 17:17