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(2) Let y = Σanz". (a) Compute y and y" and write out the first four terms of the series, as well as the coefficient of z" in the general term. (b) Show that if y"=y, then the coefficients ag and a, are arbitrary, and determine a, and as in terms of do and a. (e) Show that 12- (d) From (2c), show that a a (n+2)(n+1) do for n = 0, 1, 2, 3, ..... for n 0, 2, 4,... and da a₂ n! for n=1,3,5,....

(2) Let y = Σanz". (a) Compute y and y" and write out the first four terms of the series, as well as the coefficient of z" in the general term. (b) Show that if y"=y, then the coefficients ag and a, are arbitrary, and determine a, and as in terms of do and a. (e) Show that 12- (d) From (2c), show that a a (n+2)(n+1) do for n = 0, 1, 2, 3, ..... for n 0, 2, 4,... and da a₂ n! for n=1,3,5,....

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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HW9P2

(2) Let \( y = \sum_{n=0}^{\infty} a_n x^n \). (a) Compute \( y' \) and \( y'' \) and write out the first four terms of the series, as well as the coefficient of \( x^n \) in the general term. (b) Show that if \( y'' = y \), then the coefficients \( a_0 \) and \( a_1 \) are arbitrary, and determine \( a_2 \) and \( a_3 \) in terms of \( a_0 \) and \( a_1 \). (c) Show that \( a_{n+2} = \frac{a_n}{(n+2)(n+1)} \), for \( n = 0, 1, 2, 3, \ldots \). (d) From (2c), show that \( a_n = \frac{a_0}{n!} \) for \( n = 0, 2, 4, \ldots \) and \( a_n = \frac{a_1}{n!} \) for \( n = 1, 3, 5, \ldots \).
Transcribed Image Text:(2) Let \( y = \sum_{n=0}^{\infty} a_n x^n \). (a) Compute \( y' \) and \( y'' \) and write out the first four terms of the series, as well as the coefficient of \( x^n \) in the general term. (b) Show that if \( y'' = y \), then the coefficients \( a_0 \) and \( a_1 \) are arbitrary, and determine \( a_2 \) and \( a_3 \) in terms of \( a_0 \) and \( a_1 \). (c) Show that \( a_{n+2} = \frac{a_n}{(n+2)(n+1)} \), for \( n = 0, 1, 2, 3, \ldots \). (d) From (2c), show that \( a_n = \frac{a_0}{n!} \) for \( n = 0, 2, 4, \ldots \) and \( a_n = \frac{a_1}{n!} \) for \( n = 1, 3, 5, \ldots \).
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