Math. 350-01, Fall, 2002 Final Exam Answers, Problem 2

sol  =  
      DSolve[{y'[t]  +  (1/t)*y[t]  ==  t,  y[1] == 1}, y[t], t]

Math. 350-01, Fall, 2002 Final Exam Answers, Problem 3

sol  =  
      DSolve[{y''[t]  +  25*y[t]  == Sin[5  t],  y[0] == 1,  
          y'[0] == 0}, y[t], t]

Math. 350-01, Fall, 2002 Final Exam Answers, Problem 4

S  =  Series[y[t],  {t, 0, 4}]

solution  =  
      Solve[{D[S, {t, 2}]  +  t*D[S, {t, 1}]  ==  Sin[t],  y[0] == 0,  
          y'[0] == 0}]

Math. 350-01, Fall, 2002 Final Exam Answers, Problem 5

<< Calculus`LaplaceTransform`

Y[s_]  =  LaplaceTransform[y[t],  t, s]

Yp[s_]  =  LaplaceTransform[y'[t], t, s]

solution  =  
      Solve[{Yp[s]  + Y[s]  ==  LaplaceTransform[Sin[t], t, s],  
          y[0] == 1},  Y[s]]

y[t]  =  
      InverseLaplaceTransform[Y[s]  /.  First[solution],  s,  t]