beam2.c running 
E=10, I=9, W=7, L=8, h=1, n=9 

Analytic Solution.
beam x=0, y=-0 
                       slope=0 
beam x=1, y=-0.0198495 
                       slope=-0.0340278 
beam x=2, y=-0.0583333 
                       slope=-0.0388889 
beam x=3, y=-0.0911458 
                       slope=-0.0243056 
beam x=4, y=-0.103704 
                       slope=1.92747e-20 
beam x=5, y=-0.0911458 
                       slope=0.0243056 
beam x=6, y=-0.0583333 
                       slope=0.0388889 
beam x=7, y=-0.0198495 
                       slope=0.0340278 
beam x=8, y=-0 
                       slope=3.85494e-20 

Numerical Solution.
Fourth order difference equation solution 
boundary dy/dx=0 fourth order
n=7, h=1, C={1,-4,6,-4,1}
solve A y = F for y 
                           A                            F 
 48.000 -36.000  16.000  -3.000   0.000   0.000   0.000 0
 -4.000   6.000  -4.000   1.000   0.000   0.000   0.000 -0.00972222
  1.000  -4.000   6.000  -4.000   1.000   0.000   0.000 -0.00972222
  0.000   1.000  -4.000   6.000  -4.000   1.000   0.000 -0.00972222
  0.000   0.000   1.000  -4.000   6.000  -4.000   1.000 -0.00972222
  0.000   0.000   0.000   1.000  -4.000   6.000  -4.000 -0.00972222
  0.000   0.000   0.000  -3.000  16.000 -36.000  48.000 0

fourth order difference equations are exact within
numerical accuracy when solution is fourth order.
x=0, y=0  boundary
x=1, y=-0.0198495 
x=2, y=-0.0583333 
x=3, y=-0.0911458 
x=4, y=-0.103704 
x=5, y=-0.0911458 
x=6, y=-0.0583333 
x=7, y=-0.0198495 
x=8, y=0  boundary