para = {h -> (E^(-s)), R1 -> M/L, 
        R2 ->  M/L}
para1 = {M ->  1, L ->  1, 
        w ->  1}
para2 = {UnitStep[(L/2)] ->  1, 
        UnitStep[-(L/2)] ->  0, DiracDelta[L] ->  0, 
        UnitStep[-L] ->  0}

Q 7.2    pin-pin beam with Load M at x=0   p.172

fs = PowerExpand[+ R1 - M s - R2 h^L /. para]

qs = Expand[fs/s]
qx = InverseLaplaceTransform[qs, s, x]
qxL = qx /. x ->  L /. para2

ms = qs/s;
mx = InverseLaplaceTransform[ms, s, x]
mxL = mx /. x ->  L /. para2

Plot[{qx /. para1, mx /. para1}, {x, 0, 1}, 
      PlotRange ->  {-1, 1}, 
      PlotStyle ->  {Hue[0.9], Hue[0.7]}, 
      AspectRatio ->  0.3]

ypx = InverseLaplaceTransform[-ms/s^2, s, x]

yx = θ0 x + ypx

ans = Solve[(yx /. x ->  L) == 
              0, θ0][[1]] /. para2

δx = yx /. ans

Plot[-δx /. para1, {x, 0, 1}, 
      PlotRange -> {{0, 1},  {0, 0.1}}, PlotStyle ->  Hue[0.9], 
      AspectRatio ->  0.3, GridLines ->  Automatic]

Expand[δx]

slope at x=0

D[δx, x] /. x ->  0

search the max deflection

θa = D[δx, x]
Plot[θa /. para1, {x, 0, 1}]

ans1 = 
      FindRoot[(θa /. para1) == 0, {x, 
          0.4}]
ymax = δx /. para1 /. ans1