Volterra integral equation of second type

Question

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 method of finding the resolvent kernel and 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?

Thanks in advance!

Answer 1

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: $$ \left(2t+1 \right) \left( (2t+1) y^\prime(t) + 2 y(t) - 4 \right) = 0, \qquad y(0) = 1 $$