Numerically solve an integral equation

Question

Let $F(s)$ be a cdf on $[0,\infty[$, we have the following integral equation for $k \geq 2$ a natural number and $\lambda \in ]0,1[$. I would like to (numerically) solve the following equation: $$ F(s) = F(0) + \lambda \int_0^s F(u) du + \lambda \int_0^s(1-(1-F(u))^k) e^{u-s} du $$

Answer 1

Let $$ G(s)=\int_0^s(1−(1−F(u))^k)e^{u-s}du $$ then $$ e^s(G'(s)+G(s))=(e^sG(s))'=(1−(1−F(s))^k)e^s\\ G'(s)=1-G(s)−(1−F(s))^k $$ Replacing the integral with $G$ and taking the derivative of the original equation gives $$ F'(s)=λF(s)+λG'(s) $$ Thus you get a coupled system of two differential equations that you can solve with the ODE solver of your choice.