Solution to Problem III using Mathematica.
The problem to be solved is
This equation is also easily solved in Mathematica. Enter the command
DSolve[{(1 + y[x])*y''[x] + (y'[x])^2==0,y[0]==0,y[1]==1},y[x],x]
The result is
Mathematica also says: "Inverse functions are being used by Solve, so some solutions may not be found."
The exact solution for y(0.5) is 0.5811388.
DSolve[{y''[x]==-a*Exp[b*y[x][,y[0]==0,y[1]==1},y[x],x]