Unfortunately,DSolve[]'s support for integral equations is still somewhat limited,
so don't be surprised if some things don't work yet.
We can (solve ?) these equation by Laplace Transfrom ,but Mathematica can't find Inverse Laplace Transfrom.
eqn = Derivative[1][y][t] == -(c^2*Integrate[E^(-(b*(t - \[Tau])) - (a^2*(t - \[Tau])^2)/4)*y[\[Tau]], {\[Tau], 0, t}])
sol = LaplaceTransform[eqn, t, s]
sol2 = Solve[sol, LaplaceTransform[y[t], t, s]]
y[t] == InverseLaplaceTransform[LaplaceTransform[y[t], t, s] /. sol2[[1]], s, t]
(*y[t] == Sqrt[a^2]
InverseLaplaceTransform[1/(
Sqrt[a^2] s +
c^2 E^((b + s)^2/a^2) Sqrt[\[Pi]] Erfc[(b + s)/Sqrt[a^2]]), s,
t] y[0]*)
Regards M.I.
