Explanation Given: The differential equation is x y ″ − ( x + 2 ) y ' + 2 y = 0 . Calculation: The differential equation is given as follows: x y ″ − ( x + 2 ) y ' + 2 y = 0 ...... ( 1 ) Suppose that y = ∑ n = 0 ∞ c n x n + r . Differentiate the above equation. y ' = ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 Again differentiate the above equation. y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 Substitute the value of y ′ and y ″ equation (1) in above expression, x y ″ − ( x + 2 ) y ' + 2 y = 0 x ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 − ( x + 2 ) ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 + 2 ∑ n = 0 ∞ c n x n + r = 0 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 1 − ∑ n = 0 ∞ ( n + r ) c n x n + r − 2 ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 + 2 ∑ n = 0 ∞ c n x n + r = 0 Further simplify the above equation. [ ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) ︸ k = n c n x n + r − 1 − ∑ n = 0 ∞ ( n + r ) c n x n + r ︸ k = n + 1 − 2 ∑ n = 0 ∞ ( n + r ) c n ︸ k = n x n + r − 1 + 2 ∑ n = 0 ∞ c n x n + r ︸ k = n + 1 ] = 0 [ ∑ k = 0 ∞ ( k + r ) ( k + r − 1 ) c k x k + r − 1 − ∑ k = 1 ∞ ( k − 1 + r ) c k − 1 x k + r − 1 − 2 ∑ k = 0 ∞ ( k + r ) c k x k + r − 1 + 2 ∑ k = 1 ∞ c k − 1 x k + r − 1 ] = 0 r ( r − 1 ) c 0 x r − 1 − 2 r c 0 x r − 1 + x k + r − 1 [ ∑ k = 1 ∞ ( k + r ) ( k + r − 1 ) c k − ∑ k = 1 ∞ ( k − 1 + r ) c k − 1 − 2 ∑ k = 1 ∞ ( k + r ) c k + 2 ∑ k = 1 ∞ c k − 1 ] = 0 [ r 2 − 3 r ] c 0 x r − 1 + x k + r − 1 [ ∑ k = 1 ∞ ( k + r ) ( k + r − 3 ) c k − ∑ k = 1 ∞ ( k − 3 + r ) c k − 1 ] = 0 Compare the coefficients

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

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 6, Problem 13RE

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The two solution of the differential equation xy″−(x+2)y'+2y=0 with the use of appropriate infinite series.

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

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

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The two solution of the differential equation xy″−(x+2)y'+2y=0 with the use of appropriate infinite series.

Chapter 6 Solutions