A lightweight cantilevered beam is supported at N points spaced 10 ft apart, and a weight of 500-lb is placed at the end of the beam, 10 ft from the first support, as in the figure. Let y_k be the bending moment at the kth support. Then y₁ = 5000 ft-lb. Suppose the beam is rigidly attached at the Nth support and the bending moment there is zero. In between, the moments satisfy the three-moment equation

Bending moments on a cantilevered beam.

Find the particular solution of (15) that satisfies the boundary conditions y₁ = 5000 and y_N = 0. 

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Yk+2 +4yk+1+yk = 0 for k = 1,2,..., N - 2 (15)