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

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 18E

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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 .

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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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