Explanation The given differential equation is 2 x y ″ − y ′ + 2 y = 0 (1) Consider the solution as y = ∑ n = 0 ∞ c n x n + r . Differentiating y with respect to x gives, y ′ = ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 y ″ = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 Use the values of y , y ′ , y ″ in equation (1) then, 2 x y ″ − y ′ + 2 y = 0 2 x ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) 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 ∑ n = 0 ∞ 2 ( n + r ) ( n + r − 1 ) c n x n + r − 1 − ∑ n = 0 ∞ ( n + r ) c n x n + r − 1 + ∑ n = 0 ∞ 2 c n x n + r = 0 ∑ n = 0 ∞ [ 2 ( n + r ) ( n + r − 1 ) − ( n + r ) ] c n x n + r − 1 + ∑ n = 0 ∞ 2 c n x n + r = 0 Substitute n = k in first summation and n = k − 1 in the second summation, and then simplify ∑ k = 0 ∞ [ 2 ( k + r ) ( k + r − 1 ) − ( k + r ) ] c k x k + r − 1 + ∑ k = 1 ∞ 2 c k − 1 x k + r − 1 = 0 [ 2 ( r ) ( r − 1 ) − r ] c 0 x r − 1 + ∑ k = 1 ∞ [ 2 ( k + r ) ( k + r − 1 ) − ( k + r ) ] c k x k + r − 1 + ∑ k = 1 ∞ 2 c k − 1 x k + r − 1 = 0 [ 2 r 2 − 3 r ] c 0 x r − 1 + ∑ k = 1 ∞ [ 2 ( k + r ) ( k + r − 1 ) c k − ( k + r ) c k + 2 c k − 1 ] x k + r − 1 = 0 Thus, it gives 2 r 2 − 3 r = 0 and 2 ( k + r ) ( k + r − 1 ) c k − ( k + r ) c k + 2 c k − 1 = 0 r ( 2 r − 3 ) = 0 and ( k + r ) ( 2 k + 2 r − 3 ) c k + 2 c k − 1 = 0 On solving it gives the value of indicial roots r = 0 , 3 2

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

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ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 6.3, Problem 15E

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To show: That the indicial roots do not differ by an integer and then find the linearly independent series solution about x=0.

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That the indicial roots do not differ by an integer and then find the linearly independent series solution about x = 0 .

Solution Summary: The author shows that the indicial roots do not differ by an integer and then finds the linearly independent series solution about x=0.

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To show: That the indicial roots do not differ by an integer and then find the linearly independent series solution about x=0.

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