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(a) Suppose the power series an" converges to a function y=y(x) such that n=0 y"-y + y = 0, y (0) = 1, y (0) = 0. Find a formula that expresses an+2 in terms of an+1 and an and determine ao, a1, 02, 03. (b) Solve the initial-value problem in (a) exactly (find a simple formula for y=y(x)). (c) Use your answer in (b), Taylor series, and multiplication of power series to recover the values of ao, a1, a2, as you found in (a).

(a) Suppose the power series an" converges to a function y=y(x) such that n=0 y"-y + y = 0, y (0) = 1, y (0) = 0. Find a formula that expresses an+2 in terms of an+1 and an and determine ao, a1, 02, 03. (b) Solve the initial-value problem in (a) exactly (find a simple formula for y=y(x)). (c) Use your answer in (b), Taylor series, and multiplication of power series to recover the values of ao, a1, a2, as you found in (a).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Suppose the power series an" converges to a function y=y(x) such that n=0 y"-y + y = 0, y (0) = 1, y' (0) = 0. Find a formula that expresses an+2 in terms of an+1 and an and determine ao, a1, 02, 03. (b) Solve the initial-value problem in (a) exactly (find a simple formula for y=y(x)). (c) Use your answer in (b), Taylor series, and multiplication of power series to recover the values of ao, a1, 02, 03 you found in (a).
Transcribed Image Text:(a) Suppose the power series an" converges to a function y=y(x) such that n=0 y"-y + y = 0, y (0) = 1, y' (0) = 0. Find a formula that expresses an+2 in terms of an+1 and an and determine ao, a1, 02, 03. (b) Solve the initial-value problem in (a) exactly (find a simple formula for y=y(x)). (c) Use your answer in (b), Taylor series, and multiplication of power series to recover the values of ao, a1, 02, 03 you found in (a).
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