In[1]:=s = NDSolve[{134708*y[x] ==
y''[x]/(1 + (y'[x])^2)^1.5 + y'[x]/(1 + (y'[x])^2)^0.5,
y'[0] == 0,y'[0.005] == Cot[58*\[Pi]/180]},y,{x,0,0.005}]
Plot[Evaluate[y[x] /.s],{x,0,0.005},PlotRange -> All]
esol = Block[{\[Epsilon] = $MachineEpsilon},
NDSolve[{134708*y[x] ==
y''[x]/(1 + (y'[x])^2)^1.5 + y'[x]/(x*(1 + (y'[x])^2)^0.5),
y'[\[Epsilon]] == 0,y'[0.005] == Cot[58*\[Pi]/180]},
y,{x,\[Epsilon],0.005}]]
Plot[Evaluate[y[x] /.esol],{x,0.00001,0.005},PlotRange -> All]