ρ[r_]  =  -(4  q/a^3) r^2 Exp[-2 r/a]

v[r_]  =  
      Integrate[ρ[x], {x, 0, r}, Assumptions -> a > 0]/r + 
        Integrate[ρ[x]/x, {x, r, Infinity}, 
          Assumptions -> a > 0]

Expand[v[r]]

Integrate[ρ[x], {x, 0, Infinity}, 
      Assumptions -> a > 0]

ϕ[r]  =  Simplify[q/r  +  v[r]]

Expand[D[D[Exp[-2 r/a] (1 + r/a), r], r]]

n  = 
      2 Pi 8.854 10^(-12)/(3 10^(-11))

C1  =  Exp[n]

C1  =  Exp[10  n]

par  =  1

sph  =  (1/x^2)/Integrate[1/x^2, {x, 1, 2}]

cyl  =  1/x/Integrate[1/x, {x, 1, 2}]

par1  =  
      Piecewise[{{0, x < 1}, {par, 1 < x < 2}, {0, 2 < x}}]

sph1  =  
      Piecewise[{{0, x < 1}, {sph, 1 < x < 2}, {0, 2 < x}}]

cyl1  =  
      Piecewise[{{0, x < 1}, {cyl, 1 < x < 2}, {0, 2 < x}}]

Plot[{par1, sph1, cyl1}, {x, 0, 3}, 
      PlotStyle -> {{Thickness[0.007], 
            RGBColor[1, 0, 0]}, {Thickness[0.007], 
            RGBColor[0, 0, 1]}, {Thickness[0.007], 
            RGBColor[0, 0, 0]}}]

Export["C:\Documents and Settings\johnson\My \
Documents\Filexfer\Class06S\prob1_8.eps", %57]

cas  =  
      Table[Piecewise[{{x^(i + 1), x < 1}, {1/x^2, 
              x ≥  1}}], {i, -2, 2}]

Plot[{cas[[1]], cas[[2]], cas[[3]], cas[[4]], 
        cas[[5]]}, {x, 0, 2.5}, PlotRange ->  {0, 2}, 
      PlotStyle -> {{Thickness[0.01], Hue[0.3]}, {Thickness[0.01], 
            Hue[0.5]}, {Thickness[0.01], Hue[0.7]}, {Thickness[0.01], 
            Hue[0.9]}, {Thickness[0.01]}}, 
      AxesLabel ->  {"r", "E(r) for n=-2..2"}]

Export["C:\Documents and Settings\johnson\My \
Documents\Filexfer\Class06S\prob1_4.eps", %76]