Explanation The given differential equation is, 2 x y ″ + y ' + y = 0 ...... ( 1 ) Suppose y = ∑ n = 0 ∞ c n x n + r . Differentiate the above expression as follows: y ′ = ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 Again differentiate the above expression. y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 Substitute the value y ′ and y ″ in the equation (1). 2 x y ″ + y ' + y = 0 2 x ∑ n = 0 ∞ c n ( n + r ) ( n + r − 1 ) x n + r − 2 + ∑ n = 0 ∞ c n ( n + r ) x n + r − 1 + ∑ n = 0 ∞ c n x n + r = 0 2 ∑ n = 0 ∞ c n ( n + r ) ( n + r − 1 ) x n + r − 1 + ∑ n = 0 ∞ c n ( n + r ) x n + r − 1 + ∑ n = 0 ∞ c n x n + r = 0 { 2 r ( r − 1 ) c 0 x r − 1 + 2 ∑ n = 1 ∞ ( n + r ) ( n + r − 1 ) c n ︸ k = n x n + r − 1 + r c 0 x r − 1 + ∑ n = 1 ∞ c n ( n + r ) x n + r − 1 ︸ k = n + ∑ n = 0 ∞ c n x n + r ︸ k = n + 1 = 0 } Further simplify the above equation as follows: [ 2 r ( r − 1 ) + r ] c 0 x r − 1 + 2 ∑ k = 1 ∞ ( k + r ) ( k + r − 1 ) c k x k + r − 1 + ∑ k = 1 ∞ c k ( k + r ) x k + r − 1 + ∑ k = 1 ∞ c k − 1 x k + r − 1 = 0 ( 2 r 2 − r ) c 0 x r − 1 + ∑ k = 1 ∞ [ 2 ( k + r ) ( k + r − 1 ) c k + ( k + r ) c k + c k − 1 ] x k + r − 1 = 0 Compare the coefficients. 2 r 2 − r = 0 r ( 2 r − 1 ) = 0 r = 0 or r = 1 2 The recurrence relation is obtained as follows: 2 ( k + r ) ( k + r − 1 ) c k + ( k + r ) c k + c k − 1 = 0 [ 2 ( k + r ) ( k + r − 1 ) + 1 ] c k + c k − 1 = 0 ( k + r ) [ 2 ( k + r − 1 ) + 1 ] c k + c k − 1 = 0 ( k + r ) ( 2 k + 2 r − 1 ) c k + c k − 1 = 0 Therefore, the identical roots are r = 0 and r = 1 2

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781337515573

Author: ZILL

Publisher: Cengage

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Chapter 6, Problem 9RE

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The two solutions of the differential equation 2xy″+y'+y=0 with the use of appropriate infinite series.

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The two solutions of the differential equation 2 x y ″ + y ' + y = 0 with the use of appropriate infinite series.

Solution Summary: The author explains the two linearly independent solutions of the differential equation 2xy′′+y

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The two solutions of the differential equation 2xy″+y'+y=0 with the use of appropriate infinite series.

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