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I have an ordinary integro-differential equation of the form $$y''(t)+C_1y(t)+C_2\int_{0}^{t}f(t-\tau)y''(\tau)d\tau +C_3\int_{0}^{t}f(t-\tau)y(\tau)d\tau=0$$ where $C_1$,$C_2$ and $C_3$ are constants.
I know that finding an analytical solution can be hard. How can I solve this numerically?
Any suggestion would be greatly appreciated.
You can try to look for an analytic solution solution using Laplace transforms or using power series. $$y(t)=\sum_{n=0}^{\infty}a_nt^n$$
After your edit Laplace transform seems good as you can solve this equation as a third order ODE for : $$Y(t)=\int_{0}^{t}y(\tau)d\tau$$ And you should have noticed that : $$\int_{0}^{t}y''(\tau)d\tau=y'(t)-y'(0)=Y''(t)-Y''(0)$$ Thus your equation becomes : $$Y'''(t)+C_1Y'(t)+C_2(Y''(t)-Y''(0))+C_3Y(t)=0$$
