Chapter 9.5, Problem 13E

Consider the boundary-value problem y″ + xy = 0, y′(0) = 1, y(l) = −1.

1. (a) Find the difference equation corresponding to the differential equation. Show that for i = 0,1,2,..., n − 1 the difference equation yields n equations in n + 1 unknowns y₋₁, y₀, y₁, y₂,..., y_n−1. Here y₋₁ and y₀ are unknowns, since y₋₁ represents an approximation to y at the exterior point x = −h and y₀ is not specified at x = 0.
2. (b) Use the central difference approximation (5) to show that y₁ – y₋₁ = 2h. Use this equation to eliminate y₋₁ from the system in part (a).
3. (c) Use n = 5 and the system of equations found in parts (a) and (b) to approximate the solution of the original boundary-value problem.