Can Mathematica solve integro-differential equations?

Question

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.

Answer 1

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]).