Explanation Procedure used: Check x = x 0 is an ordinary point. Write y in the form of powers of x . y = ∑ n = 0 ∞ c n x n Write y ′ in the form of powers of x . y ′ = ∑ n = 1 ∞ n c n x n − 1 Write y ″ in the form of powers of x . y ″ = ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 Substitute values of y , y ′ and y ″ in given differential equation. Equate the like powers of x . Find values of constant and substitute in y . Calculation: Consider the differential equation. y ″ − x y ′ + 4 y = 0 … ( 1 ) Since, the point x = 0 is an ordinary point for the given differential equation as, p ( 0 ) = 0 and q ( 0 ) = 4 . So, assume ∑ n = 0 ∞ c n x n be the series solution for the given differential equation. Substitute ∑ n = 0 ∞ c n x n for y , ∑ n = 1 ∞ n c n x n − 1 for y ′ and ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 for y ″ in equation ( 1 ) . ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 − x ∑ n = 1 ∞ n c n x n − 1 + 4 ∑ n = 0 ∞ c n x n = 0 ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 − ∑ n = 1 ∞ n c n x n + 4 ∑ n = 0 ∞ c n x n = 0 Substitute n in place of ( n − 2 ) in the first term of the above equation. ∑ n = 0 ∞ ( n + 2 ) ( n + 1 ) c n + 2 x n − ∑ n = 1 ∞ n c n x n + 4 ∑ n = 0 ∞ c n x n = 0 ( 2 c 2 + 4 c 0 ) + ∑ n = 1 ∞ [ ( n + 2 ) ( n + 1 ) c n + 2 − n c n + 4 c n ] x n = 0 Equate coefficient of power of x

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

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Chapter 4.8, Problem 12E

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The solution of differential equation y″−xy′+4y=0 with power series expansion about x=0.

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Chapter 4.8, Problem 11E

Chapter 4.8, Problem 13E

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Chapter 4 Solutions

Introductory Differential Equations

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The solution of differential equation y ″ − x y ′ + 4 y = 0 with power series expansion about x = 0 .

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The solution of differential equation y″−xy′+4y=0 with power series expansion about x=0.

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Chapter 4 Solutions