(* Example 1 *)
Simplify[DSolve[{x'[t]==2 x[t]+y[t],y'[t]==3 x[t], 
                 x[0]==2,y[0]==-2},
                 {x[t],y[t]},t]]
                 

Simplify[MatrixExp[t {{2,1},{ 3,0}}].{2,-2}]                 

(* Example 2 *)

Simplify[DSolve[{x'[t]==x[t]+y[t],y'[t]==-x[t]+y[t],
                 x[0]==2,y[0]==-2},
                 {x[t],y[t]},t]]                 

Simplify[MatrixExp[t {{1,1},{-1,1}}].{2,-2}]

(* Example 3  *)

Simplify[DSolve[{x'[t]==-2x[t]+y[t],y'[t]==-x[t],
                 x[0]==2,y[0]==-2},
                 {x[t],y[t]},t]]

Simplify[MatrixExp[t {{-2,1},{-1,0}}].{2,-2}]

