I have a set of coupled integro-differential equations: $$ \frac{dx_i(t)}{dt}=-x_i(t)+f_i(\mathbf{x}(t))+\sum_j{\partial_{x_j}f_i(\mathbf{x}(t))\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')} }$$
The integrals, $ \psi_j(t) =\int_{0}^{t}dt'f_j(\mathbf{x}(t'))e^{-(t-t')} $, are solutions of the differential equations $$ \frac{d\psi_j(t)}{dt}+\psi_j(t)=f_j(\mathbf{x}(t)) $$ with $\psi(0)=0$. I've been told by my supervisor that the numerical evaluation of the coupled system can be greatly sped up by using these differential equations but I don't understand how - does anyone have any ideas? I get the feeling I'm missing something completely obvious.