y[t]  =  α * t * Cos[t]  +  β * t * Sin[t]

v  =  Simplify[D[D[y[t], t], t]  +  y[t]]

D[D[-1/2 * t*Cos[t], t], t]  -  1/2*t*Cos[t]

sol1  =  
      DSolve[{Q''[t]  +  .01*Q'[t]  +  Q[t]  ==  Sin[t], 
          Q[0] == 0, Q'[0] == 0}, Q[t], t]

Q1[t_]  =  Simplify[Q[t]  /.  First[sol1]]

Q1[0]

sol2 =  
      DSolve[{Q''[t]  +  Q[t]  ==  Sin[t], Q[0] == 0, 
          Q'[0] == 0}, Q[t], t]

Q2[t_]  =  Simplify[Q[t]  /.  First[sol2]]

ub  =  200

Plot[Q1[t],  {t, 0, ub}]

Plot[Q2[t], {t, 0, ub}]