example 5.1  p.61

ans 1

LaplaceTransform[17 Sin[w t], t, 
      s]   
Simplify[
      InverseLaplaceTransform[%/(  s + 1), s, t] /. 
        w ->  4]

yp = -4 Cos[4 t] + Sin[4 t]

    D[ yp, t] + yp

yt = c1 Exp[-t]+yp

Solve[(yt /. t ->  0) == 50, 
        c1][[1]]
ytt = yt /. %

Plot[{ytt, yp}, {t, 0, 6}, AspectRatio ->  0.3, 
      PlotRange ->  {-5, 50}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}]

ans 2

eq = y'[t] + y[t] == 17 Sin[w t] /. 
        w ->  4

DSolve[{eq, y[0] == 50}, y[t], t]