Clear[v, v1, v2, v3, c1, c2, c3, c4]
  
v3 = 
      Integrate[(w + P DiracDelta[x - d])/EI, x] + 
        c1
v2 = Integrate[v3, x] + c2
v1 = Integrate[v2, x] + c3
v = Integrate[v1, x] + c4

v2 /. x -> 0 /. UnitStep[L - d] -> 1 /. 
      UnitStep[-d] -> 0
v /. x -> 0 /. UnitStep[L - d] -> 1 /. 
      UnitStep[-d] -> 0

v = v /. c2 -> 0 /. 
          c4 -> 0;
v2 = v2 /. c2 -> 0 /. 
          c4 -> 0;
eq2 = v2 /. x -> L /. UnitStep[L - d] -> 1 /. 
        UnitStep[-d] -> 0
v = 
      v /. Solve[eq2 == 0, c1][[1]]
v2 = v2 /. Solve[eq2 == 0, c1][[1]]

eq = 
      Simplify[v /. x -> L /. UnitStep[L - d] -> 1 /. 
              UnitStep[-d] -> 0 /. c2 -> 0 /. 
          c4 -> 0]
v = 
      v /. Solve[eq == 0, c3][[1]]
v2 = v2 /. Solve[eq == 0, c3][[1]]

valuep = {EI -> 1, L -> 1, d -> 0.5, P -> 1, 
          w -> 0};
valuew = {EI -> 1, L -> 1, d -> 0.5, P -> 0, 
          w -> 1};
nvp = v /. valuep
nv2p = v2 /. valuep
nvw = v /. valuew
nv2w = v2 /. valuew
Plot[{nvp, nvw}, {x, 0, 1}, 
      PlotStyle -> {Hue[0.2], Hue[1]}]
{nvp /. x -> 0.5, nvw /. x -> 0.5, nvp /. x -> 0.25, 
      nvw /. x -> 0.25}
Plot[{-nv2p, -nv2w}, {x, 0, 1}, 
      PlotStyle -> {Hue[0.2], Hue[1]}]
{-nv2p /. x -> 0.5, -nv2w /. x -> 0.5, -nv2p /. 
        x -> 0.25, -nv2w /. x -> 0.25}