Explanation Given: The given differential equation is, x 3 y ″ − x 2 y ′ − y = 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 3 y ″ − x 2 y ′ − y = 0 Change the variable of the differential equation bye, z = 1 x y ′ = d y d z d z d x = d y d z ( − 1 x 2 ) = − z 2 d y d z y ″ = d d x ( − 1 x 2 d y d z ) = − 1 x 2 d 2 y d z 2 ( − 1 x 2 ) − d y d z ( − 2 x 3 ) = z 4 d 2 y d z 2 + 2 z 3 d y d z Substitute the values of y ′ and y ″ in the given differential equation to get, x 3 ( z 4 d 2 y d z 2 + 2 z 3 d y d z ) + x 2 ( − z 2 d y d z ) − y = 0 Substitute 1 z for x to get, 1 z 3 ( z 4 d 2 y d z 2 + 2 z 3 d y d z ) − 1 z 2 ( − z 2 d y d z ) − y = 0 z d 2 y d z 2 + 2 d y d z + d y d z − y = 0 z d 2 y d z 2 + 3 d y d z − y = 0 d 2 y d z 2 + 3 z − 1 d y d z − z − 1 y = 0 … ( 4 ) So, p ( z ) = 3 z − 1 and q ( z ) = − z − 1 p 0 = lim z → 0 z p ( z ) = lim z → 0 z 3 z − 1 = 3

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 8.6, Problem 41E

To determine

To show:

Infinity is a singular point of the given differential equation and find the first nonzero terms in the series expansion of the solution about infinity.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.6, Problem 40E

Chapter 8.6, Problem 42E

Students have asked these similar questions

Can you cut the 12 glass triangles from a sheet of glass that is 4 feet by 8 feet? If so, how can it be done?

Can you cut 12 glass triangles from a sheet of glass that is 4 feet by 8 feet? If so, draw a diagram of how it can be done.

Use the method of undetermined coefficients to solve the given nonhomogeneous system. x-()*+(5) = 1 3 3 1 X+ t +3 -1 -2t 1 x(t) = º1 1 e +021 e +

Chapter 8 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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.

Infinity is a singular point of the given differential equation and find the first nonzero terms in the series expansion of the solution about infinity.

Solution Summary: The author explains how to derive a series solution about the singular point of the given differential equation about infinity.

Concept explainers

Videos

To show: Infinity is a singular point of the given differential equation and find the first nonzero terms in the series expansion of the solution about infinity.

Want to see the full answer?

Chapter 8 Solutions

To show:

Infinity is a singular point of the given differential equation and find the first nonzero terms in the series expansion of the solution about infinity.