Explanation Given: The given differential equation is, ( x + 2 ) x 2 y ″ ( x ) − x y ′ ( x ) + ( 1 + x ) y ( x ) = 0 , x > 0 . And, w ( r , x ) = x r ∑ n = 0 ∞ a n x n = ∑ n = 0 ∞ a n x n + r is a solution of the differential equation. The given recurrence relation is, ( k + r − 1 ) ( 2 k + 2 r − 1 ) a k + [ ( k + r − 1 ) ( k + r − 2 ) + 1 ] a k − 1 = 0 . Approach: To derive a series solution about the singular point x 0 of the equation a 2 ( x ) y ″ ( x ) + a 1 ( x ) y ′ ( x ) + a 0 ( x ) y ( x ) = 0 , x > x 0 … ( 1 ) The following steps are given below. 1. Set p ( x ) = a 1 ( x ) a 2 ( x ) , q ( x ) = a 0 ( x ) a 2 ( x ) . If both ( x − x 0 ) p ( x ) and ( x − x 0 ) 2 q ( x ) are analytic at x 0 , then x 0 is a regular point and remaining steps apply. 2. Let w ( r , x ) = ( x − x 0 ) r ∑ n = 0 ∞ a n ( x − x 0 ) n = ∑ n = 0 ∞ a n ( x − x 0 ) n + r … ( 2 ) And, using term wise differentiation, substitute w ( r , x ) into equation ( 1 ) to obtain an equation of the form A 0 ( x − x 0 ) r + J + A 1 ( x − x 0 ) r + J + 1 + ... = 0 3. Set the coefficients A 0 , A 1 , A 3 , equal to zero. [Notice that the equation A 0 = 0 is just a constant multiple of the indicial equation r ( r − 1 ) + p 0 r + q 0 = 0 here, p 0 = lim x → 0 x p ( x ) , q 0 = lim x → 0 x 2 q ( x ) , and r 0 = lim x → 0 x 3 r ( x ) .] 4. Use the system of equations A 0 = 0 , A 1 = 0 , ... , A k = 0 to find a recurrence relation involving a k and a 0 , a 1 ,…, a k − 1 . 5. Take r = r 1 , the larger root of the indicial equation, and use the obtained in step 4 to determine a 1 , a 2 ,…, recursively in terms of a 0 and r 1 . 6. A series expansion of a solution to equation ( 1 ) is, w ( r 1 , x ) = ( x − x 0 ) r 1 ∑ n = 0 ∞ a n ( x − x 0 ) n , x > x 0 … ( 3 ) . Calculation: Consider the given differential equation, ( x + 2 ) x 2 y ″ ( x ) − x y ′ ( x ) + ( 1 + x ) y ( x ) = 0 And, w ( r , x ) = x r ∑ n = 0 ∞ a n x n = ∑ n = 0 ∞ a n x n + r . So, w ′ ( r , x ) = ∑ n = 0 ∞ ( n + r ) a n x n + r − 1 w ″ ( r , x ) = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) a n x n + r − 2 r satisfies the indicial equation defined as, To determine (b) To find: The second series solution with r = 1 2 in the recurrence relation in part a. To determine (c) To find: w ( 1 , x ) using the recurrence relation in part a with r = 1 .

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.6, Problem 44E

To determine

(a)

To show:

Coefficients ak k≥1 of the series solution of the given differential equation, satisfy the given recurrence relation.

To determine

(b)

To find:

The second series solution with r=12 in the recurrence relation in part a.

To determine

(c)

To find:

w(1,x) using the recurrence relation in part a with r=1.

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Chapter 8.6, Problem 43E

Chapter 8.6, Problem 45E

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(a) To show: Coefficients a k k ≥ 1 of the series solution of the given differential equation, satisfy the given recurrence relation.

Solution Summary: The author explains how the series solution of the given differential equation, a_k, satisfy the recurrence relation.

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(a)

To show:

Coefficients ak k≥1 of the series solution of the given differential equation, satisfy the given recurrence relation.

(b)

To find:

The second series solution with r=12 in the recurrence relation in part a.

(c)

To find:

w(1,x) using the recurrence relation in part a with r=1.

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Chapter 8 Solutions

(a) To show: Coefficients a k k ≥ 1 of the series solution of the given differential equation, satisfy the given recurrence relation.

Solution Summary: The author explains how the series solution of the given differential equation, a_k, satisfy the recurrence relation.

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(a) To show: Coefficients ak k≥1 of the series solution of the given differential equation, satisfy the given recurrence relation.

(b) To find: The second series solution with r=12 in the recurrence relation in part a.

(c) To find: w(1,x) using the recurrence relation in part a with r=1.

Want to see the full answer?

Chapter 8 Solutions

(a)

To show:

Coefficients ak k≥1 of the series solution of the given differential equation, satisfy the given recurrence relation.

(b)

To find:

The second series solution with r=12 in the recurrence relation in part a.

(c)

To find:

w(1,x) using the recurrence relation in part a with r=1.