Explanation Given: Given differential equation is y ″ − 2 y ′ + y = 0 . Approach: To write general solution of a differential equation, p ( x ) and q ( x ) must be analytic at a point x = a , then x = a would be an ordinary point and the given differential equation can be expressed in general solution in the form y ( x ) = ∑ n = 0 ∞ a n x n . Calculation: Consider the given differential equation. y ″ − 2 y ′ + y = 0 …… (1) Let the general solution of given equation is, y ( x ) = ∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + .... Differentiate above equation upto second derivative. y ′ ( x ) = ∑ n = 1 ∞ n a n x n − 1 y ″ ( x ) = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 Substitute value of y ( x ) , y ′ ( x ) and y ″ ( x ) in equation (1). ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + 2 ∑ n = 1 ∞ n a n x n − 1 + ∑ n = 0 ∞ a n x n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + 2 ∑ n = 1 ∞ n a n x n − 1 + ∑ n = 0 ∞ a n x n = 0 …… ( 2 ) Assume k = n − 2 for first and k = n − 1 for second summation. ∑ k = 0 ∞ ( k + 1 ) ( k + 2 ) a k + 2 x k − 2 ∑ k = 0 ∞ ( k + 1 ) a k + 1 x k + ∑ k = 0 ∞ a k x k = 0 ∑ k = 0 ∞ [ ( k + 1 ) ( k + 2 ) a k + 2 − 2 ( k + 1 ) a k + 1 + a k ] x k = 0 Compare the constant term and coefficient of x n in above equation. ( k + 1 ) ( k + 2 ) a k + 2 − 2 ( k + 1 ) a k + 1 + a k = 0 ( k + 1 ) ( k + 2 ) a k + 2 = 2 ( k + 1 ) a k + 1 − a k a k + 2 = 2 ( k + 1 ) a k + 1 − a k ( k + 1 ) ( k + 2 ) …… ( 4 ) Substitute 0 for k in equation (4)

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.3, Problem 16E

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The first four nonzero terms of general solution of a differential equation obtained about x=0.

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Chapter 8.3, Problem 15E

Chapter 8.3, Problem 17E

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The first four nonzero terms of general solution of a differential equation obtained about x = 0 .

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The first four nonzero terms of general solution of a differential equation obtained about x = 0 .

Solution Summary: The author explains that the first four nonzero terms of general solution of a differential equation obtained about x=0.

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To find: The first four nonzero terms of general solution of a differential equation obtained about x=0.

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Chapter 8 Solutions

To find:

The first four nonzero terms of general solution of a differential equation obtained about x=0.