Q 5.6 (1)   y'[t]+y[t]==17 Sin[w t]/.w-> 4  y[0]=50   p.99

fs = LaplaceTransform[17 Sin[4 t], t, s]

ys = Simplify[Factor[(y0 + fs)/(s + 1)]]

yt = 
      Simplify[InverseLaplaceTransform[ys /. y0 ->  50, s, 
          t]]

yt2 = 
      Simplify[InverseLaplaceTransform[ys /. y0 -> 0, s, t]]

Plot[{yt, yt2}, {t, 0, 6}, AspectRatio ->  0.3, 
      PlotRange ->  {-5, 50}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}]

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

D[yp, t] + yp

Plot[{yt, yt2, yp}, {t, 0, 6}, AspectRatio ->  0.3, 
      PlotRange ->  {-5, 50}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7], Hue[0.5]}]

NDSolve  別解 p.99

eq=y[t]+y'[t]==17 Sin[4 t]

ans = ExpandAll[
          DSolve[{eq, y[0] == 50}, y[t], 
            t]][[1]]
yta = y[t] /. ans;
yta - yt

Plot[{yta, yp}, {t, 0, 6}, AspectRatio ->  0.3, 
      PlotRange ->  {-5, 50}, 
      PlotStyle ->  {Hue[0.8], Hue[0.7]}]