Consider the boundary-value problem (a) Find the

Chapter 9, Problem 13E

(choose chapter or problem)

Consider the boundary-value problem \(y^{\prime \prime}+x y=0\), \(y^{\prime}(0)=1\), y(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 unknows \(y_{-1}, y_{0}, y_{1}, y_{2}, \ldots, y_{n-1}\). Here \(y_{-1}\) and \(y_{0}\) are unknowns, since \(y_{1}\) represents an approximation to y at the exterior point x = -h and \(y_{0}\) is not specified at x = 0.

(b) Use the central difference approximation (5) to show that \(y_{1}-y_{-1}=2 h\). Use this equation to eliminate \(y_{-1}\) from the system in part (a).

(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.

Text Transcription:

y^prime prime+x y=0

y^prime 0=1

y_2

y_n-1

y_-1

y_0

y_1-y_-1=2 h