# Can Mathematica solve integro-differential equations?

I have integro-differential equations like this:

γ = 0.1;
κ = 0.15;
g = 0.2;
δ = 0.2 + 0.6 I;

eqns = {
x'[t] == -γ x[t] - g Re@z[t],
y'[t] == -κ y[t] + g Re@z[t],
z[t] ==
Integrate[(x[τ] -
y[τ]) Exp[ -δ (t - τ)], {τ, 0, t}]
};

ints = {
x[0] == 1,
y[0] == 0
};

NDSolve[Join[eqns, ints], {x, y}, {t, 0, 10}]


I don't know how to use Mathematica to solve it or if it can be solved at all using some combinations of built-in functions?

To solve integro-differential equations in Mathematica is important to me for studying some special physical models.

• – Michael E2 Oct 10 '13 at 13:00
• Solving integral equations is hard enough. In general there is no systematic approach. Look e.g. here: How to solve system of integral equations how one can get a general idea of possible solutions. There is no built-in functionality in any computer system for solving inegro-differential equations as far as I can say. – Artes Oct 10 '13 at 13:20
• Also might want to check responses to similar question here – Daniel Lichtblau Oct 10 '13 at 18:28

For your special problem, it seems you can differentiate the third equation and transform it into the differential equation x[t] - y[t] - δ z[t]==z'[t]. You can also deduce boundary conditions from the integral equation (compute z[0]).