(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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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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