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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.

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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
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1 Answer

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]).

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Hi Ahmed, welcome to Mathematica StackExchange! Don't forget to upvote good answers (and other people's questions) using the triangle above the number next to the post. I edited the formatting of your answer as per the guide found on the help centre. It's a good idea to read the guide and the about page –  gpap Oct 10 '13 at 20:41
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