DSolve returns unevaluated(I need a closed or numerical solution for this system and plot solution)

Question

I am trying to find a solution to this system of differential equations but the program gives the same output without any messages. I would like help. Please find the Mathematica code posted here. a and d are Constants, $I=\sqrt{-1}$

S1 = I y' == Cos[a t] Exp[- I d t] x;

S2 = I x' == Cos[a t] Exp[ I d t] y;

S = {S1, S2};

DSolve[S, {y, x}, t]

Answer 1

I haven't found a way to get DSolve to crack it, so turn to NDSolve

a=1;d=2;
sol={y[t],x[t]}/.NDSolve[{
  I y'[t] == Cos[a t] Exp[-I d t] x[t], I x'[t] == Cos[a t] Exp[I d t] y[t],
  y[0]==1,x[0]==1}, {y[t], x[t]}, {t,0,2}][[1]];
Plot[{Re[sol[[1]]],Im[sol[[1]]],Re[sol[[2]]],Im[sol[[2]]]},{t,0,2}]

which instantly gives you plots of the real and complex components of x and y.