Excuse me in advance in case of inaccuracies, I am absolutely not familiar with such a type of integro-differential equations. Let us consider the following equation $$ \frac{d^2 y}{dt^2} = g(t) + \lambda \int\limits_{- \infty}^{t} F\left(t, t', y(t), y(t'), \frac{dy(t)}{dt}, \frac{dy(t')}{dt}\right) dt', $$ where $\lambda \in \mathrm{R}$, and $g$ is a smooth function. There also known a solution for $t<0$ which reads: $y(t) = h(t)$ (here $h(t)$ is a known function). The problem is to solve the second-order integro-differential equation for $t > 0$.
In my case, I do not believer it can be solved analytically. Talking about the numerical approach, I would typically act in the following way: first, approximate the integral with finite sums formulas, such as rectangular, trapezoidal, Simpson, or else; second, approximate the function $y(t)$ with finite difference method, obtain the system of linear equations and solve it by preferable method. I understand how to make something like this in the case of the integral with fixed ends, but have no idea how to implement it in the described problem.
I would be very glad for any recommendations/solutions/references (preferably, not for specialists in numerical methods, which I will be able to understand).