DSolve[{x'[t] == 
          1, y'[t] == -x[t] - 2  y[t] + 1}, {x, y}, 
      t]

x[t_] := t;
y[t_] := 
        (1/2) (E^(-2 t)) (((3 (E^(2 t)))/2) - t) - 
          (1/2) (E^(-2 t)) (-1 + (E^(2 t))) t;

(1/2) (E^(-2 t)) (((3 (E^(2 t)))/2) - t) - 
        (1/2) (E^(-2 t)) (-1 + (E^(2 t))) t   // Simplify

y'[t] + x[t] + 2  y[t] // Simplify

Integrate[(1/2) ((E^(2 t)) - 1) + (E^(2 t)), {t, 0, 
        l}]

m = {{1, 0}, {((-1)/2) (1 - (E^((-2 t)))), 
          (E^(-2 t))}}

im = Inverse[m] // Simplify

Integrate[im  .  {1, 1}, {t, 0, T}] // MatrixForm

m  .  (Out[42] /. {T -> t}) // Simplify

Out[42] /. {T -> t}

v = {t, 
        (1/4) (-3 + 3 (E^(2 t)) - 2 t)}

v // MatrixForm

m  .  v  //  Simplify  //  MatrixForm

p  =  {