Explanation The given differential equation is x 2 y ″ − ( x − 2 9 ) 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 ″ in equation (1) then, x 2 y ″ − ( x − 2 9 ) y = 0 x 2 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − 2 − ( x − 2 9 ) ∑ n = 0 ∞ c n x n + r = 0 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) c n x n + r − ∑ n = 0 ∞ c n x n + r + 1 + ∑ n = 0 ∞ 2 9 c n x n + r = 0 ∑ n = 0 ∞ [ ( n + r ) ( n + r − 1 ) + 2 9 ] c n x n + r − ∑ n = 0 ∞ c n x n + r + 1 = 0 Substitute n = k in first summation and n = k − 1 in the second summation, and then simplify ∑ k = 0 ∞ [ ( k + r ) ( k + r − 1 ) + 2 9 ] c k x k + r − ∑ k = 1 ∞ c k − 1 x k + r = 0 [ ( r ) ( r − 1 ) + 2 9 ] c 0 x r + ∑ k = 1 ∞ [ ( k + r ) ( k + r − 1 ) + 2 9 ] c k x k + r − ∑ k = 1 ∞ c k − 1 x k + r = 0 [ r 2 − r + 2 9 ] c 0 x r + ∑ k = 1 ∞ [ ( k + r ) ( k + r − 1 ) c k + 2 9 c k − c k − 1 ] c k x k + r = 0 Thus, it gives r 2 − r + 2 9 = 0 And ( k + r ) ( k + r − 1 ) c k + 2 9 c k − c k − 1 = 0 On solving it gives the value of indicial roots r = 1 3 , 2 3 . Hence the indicial roots do not differ by an integer. Now, for r = 1 3 the recurrence relation is ( k + 1 3 ) ( k + 1 3 − 1 ) c k + 2 9 c k − c k − 1 = 0 ( 3 k + 1 ) 3 ( 3 k − 2 ) 3 c k + 2 9 c k − c k − 1 = 0 c k = 3 k ( 3 k − 1 ) c k − 1 for k = 1 , 2 , 3 ,

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

11th Edition

ISBN: 9781337515573

Author: ZILL

Publisher: Cengage

See similar textbooks

Videos

Question

Chapter 6.3, Problem 20E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.3, Problem 19E

Chapter 6.3, Problem 21E

Students have asked these similar questions

Find three horizontal tangents between [0,10]

Not use ai please

No chatgpt pls will upvote

Chapter 6 Solutions

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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.

Videos

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

Want to see the full answer?

Chapter 6 Solutions