Explanation Given information: The given differential equation is y ' ' − ( x − 1 ) y − y = 0 . Concept Used: Power series is defined as an infinite series of the form, ∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ......... + a n x n + .... , where x is a variable. Calculation: Assume y = ∑ n = 0 ∞ a n ( x − 1 ) n is a solution of the differential equation. Then, differentiating the assumed solution with respect to x , y ' = ∑ n = 1 ∞ n a n ( x − 1 ) n − 1 Again differentiating above equation with respect to x , y ' ' = ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 Now, the given differential equation is, y ' ' − ( x − 1 ) y − y = 0 Inserting the values of y and y ' ' in above equation, y ' ' − ( x − 1 ) y − y = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 − ( x − 1 ) ∑ n = 0 ∞ a n ( x − 1 ) n − ∑ n = 0 ∞ a n ( x − 1 ) n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 − ∑ n = − 1 ∞ a n ( x − 1 ) n + 1 − ∑ n = 0 ∞ a n ( x − 1 ) n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 = ∑ n = − 1 ∞ a n ( x − 1 ) n + 1 + ∑ n = 0 ∞ a n ( x − 1 ) n ∑ n = 0 ∞ ( n + 2 ) ( n + 1 ) a n + 2 ( x − 1 ) n = ∑ n = 0 ∞ a n − 1 ( x − 1 ) n + ∑ n = 0 ∞ a n ( x − 1 ) n Equating the coefficients of like terms, one can obtain the recursion formula. ( n + 2 ) ( n + 1 ) a n + 2 = a n − 1 + a n Which implies that, a n + 2 = a n − 1 ( n + 2 ) ( n + 1 ) + a n ( n + 2 ) ( n + 1 ) , n ≥ 0 Above formula will generate the following results, a 0 a 1 a 2 = a 0 − 1 ( 0 + 2 ) ( 0 + 1 ) + a 0 ( 0 + 2 ) ( 0 + 1 ) = a 0 2 ! a 3 = a 1 − 1 ( 1 + 2 ) ( 1 + 1 ) + a 1 ( 1 + 2 ) ( 1 + 1 ) = a 0 3 ! + a 1 3 ! = a 0 + a 1 3 ! a 4 = a 2 − 1 ( 2 + 2

11th Edition

ISBN: 9781337604789

Author: Larson

Publisher: Cengage

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 16, Problem 15PS

To determine

To calculate: The first eight terms of the solution of the Airy’sequation, y''−(x−1)y−y=0, using a power series.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16, Problem 14PS

Chapter 16, Problem 16PS

Students have asked these similar questions

The graph of f(x) is given below. Select each true statement about the continuity of f(x) at x = 1. Select all that apply: ☐ f(x) is not continuous at x = 1 because it is not defined at x = 1. ☐ f(x) is not continuous at x = 1 because lim f(x) does not exist. x+1 ☐ f(x) is not continuous at x = 1 because lim f(x) ‡ f(1). x+→1 ☐ f(x) is continuous at x = 1.

a is done please show b

A homeware company has been approached to manufacture a cake tin in the shape of a "ghost" from the Pac-Man video game to celebrate the 45th Anniversary of the games launch. The base of the cake tin has a characteristic dimension / and is illustrated in Figure 1 below, you should assume the top and bottom of the shape can be represented by semi-circles. The vertical sides of the cake tin have a height of h. As the company's resident mathematician, you need to find the values of r and h that minimise the internal surface area of the cake tin given that the volume of the tin is Vfixed- 2r Figure 1 - Plan view of the "ghost" cake tin base. (a) Show that the Volume (V) of the cake tin as a function of r and his 2(+1)²h V = 2

Chapter 16 Solutions

Bundle: Llf Multivariable Calculus

Ch. 16.1 - Exactness What does it mean for the...Ch. 16.1 - Integrating Factor When is it beneficial to use an...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

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

The first eight terms of the solution of the Airy’sequation, y ' ' − ( x − 1 ) y − y = 0 , using a power series.

Solution Summary: The author calculates the first eight terms of the solution of Airy's equation using a power series.

Concept explainers

Videos

To calculate: The first eight terms of the solution of the Airy’sequation, y''−(x−1)y−y=0, using a power series.

Want to see the full answer?

Chapter 16 Solutions