para = h -> (E^(-s))

Q 7.1a(1)  p.171

us1 = 30/s - 30/s^2 (h - h^2) /. para

ut1 = InverseLaplaceTransform[us1, s, x]
Plot[ut1, {x, 0, 3}, AspectRatio ->  0.3, 
      PlotRange ->  {{0, 3}, {0, 30}}]

fs1 = R1 - us1 + 15 s h^2 + R2 h^3 /. para

Q 7.1a(3)

qs = Expand[fs1/s]
qx = InverseLaplaceTransform[qs, s, x]
% /. x ->  3

ms = qs/s;
mx = InverseLaplaceTransform[ms, s, x]
% /. x ->  3

ans = Solve[{(qx /. x ->  3) == 
              0, (mx /. x ->  3) == 0}, {R1, 
            R2}][[1]]

Plot[{qx /. ans, mx /. ans}, {x, 0, 3}, 
      PlotRange ->  {-20, 30}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}, 
      AspectRatio ->  0.3]

Q 7.1b(1)

us2 = 300/s^2 (1 - h - h^2 + h^3) /. para

ut2 = InverseLaplaceTransform[us2, s, x]
Plot[ut2, {x, 0, 4}, AspectRatio ->  0.3, 
      PlotRange -> {{0, 4},  {-100, 300}}]

fs2 = -50 s + R1 - us2 + 100 h^2 + 200 s h^3 + R2 h^4 /. 
        para

Q 7.1b(3)

qs = Expand[fs2/s]
qx = InverseLaplaceTransform[qs, s, x]
% /. x ->  4

ms = qs/s;
mx = InverseLaplaceTransform[ms, s, x]
% /. x ->  4

ans = Solve[{(qx /. x ->  4) == 
              0, (mx /. x ->  4) == 0}, {R1, 
            R2}][[1]]

Plot[{qx /. ans, mx /. ans}, {x, 0, 4}, 
      PlotRange ->  {-250, 300}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}, 
      AspectRatio ->  0.3]