Problems with DSolve[] funtion

Question

I'm trying to solve a differential equation but Mathematica doesn't do it. Can you Help me please? Why doesn't Mathematica do it? this is my code

DSolve[{M'[t] == -(a + I b) M[t] - I o Exp[-I b t] Z[t]
  , P'[t] == -(a - I b) M[t] + I o Exp[I b t] Z[t],
  Z'[t] == -2 a Z[t] + 2 I o Exp[-I b t] P[t] - 
    2 I O Exp[I b t] M[t]}, {M[t], P[t], Z[t]}, t]

but Mathematica doesn't do it

Im following this book and the equation and its solution

Answer 1

While I don't know why mathematica won't solve the eqn. system as it is, mathematica does solve it if you make a change of variables. You can change the eqns into a first order linear systems of ODE using M(t)=Exp[-I b t] m(t) and P(t)=Exp[I b t] p(t)

DSolve[{m'[t] - I b m[t] == -(a + I b) m[t] - I o Z[t], 
p'[t] + I b p[t] == -(a - I b) p[t] + I o Z[t], 
Z'[t] == -2 a Z[t] + 2 I o p[t] - 2 I o m[t]}, {m[t], p[t], Z[t]}, t]

works. Also I think your second eqn has a mistake, M(t) should be P(t).