Explanation Formula used: Power Series Method : The power series method for solving a second-order homogenous differential equation consists of finding the coefficients of a power series y ( x ) = ∑ n = 0 ∞ c n x n = c 0 + c 1 x + c 2 x 2 + … Calculation: The given equation is y ″ + 2 y ′ + y = 0 . Assume that the series solution takes the form of y = ∑ n = 0 ∞ c n x n . Calculate the derivative of y = ∑ n = 0 ∞ c n x n . y ′ = ∑ n = 1 ∞ n c n x n − 1 Now calculate the second derivative of y = ∑ n = 0 ∞ c n x n . y ″ = ∑ n = 2 ∞ n ( n − 1 ) c n x n − 1 − 1 = ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 Substitute the derivatives of y ″ and y ′ in y ″ + 2 y ′ + y = 0 as follows. ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 + ∑ n = 1 ∞ 2 n c n x n − 1 + ∑ n = 0 ∞ c n x n = 0 Equate the coefficient of each power of x to zero as shown in the following table. Power of x Coefficient equation x 0 2 ( 1 ) c 2 + 2 ( 1 ) c 1 + c 0 = 0 or c 2 = − c 1 − 1 2 c 0 x 1 3 ( 2 ) c 3 + 2 ( 2 ) c 2 + c 1 = 0 or c 3 = − 2 3 c 2 − 1 6 c 1 = 1 2 c 1 + 1 3 c 0 x 2 4 ( 3 ) c 4 + 2 ( 3 ) c 3 + c 2 = 0 or c 4 = − 1 2 c 3 − 1 12 c 2 = − 1 6 c 1 − 1 8 c 0 x 3 5 ( 4 ) c 5 + 2 ( 4 ) c 4 + c 3 = 0 or c 5 = − 2 5 c 4 − 1 20 c 3 = 1 24 c 1 + 1 30 c 0 x 4 6 ( 5 ) c 6 + 2 ( 5 ) c 5 + c 4 = 0 or c 6 = − 1 3 c 5 − 1 30 c 4 = − 1 120 c 1 − 1 144 c 0 x n ( n + 2 ) ( n + 1 ) c n + 2 + 2 ( n + 1 ) c n + 1 + c n = 0 or c n + 2 = − 2 n + 2 c n + 1 − 1 ( n + 2 ) ( n + 1 ) c n The recursive relation is c n + 2 = − 2 n + 2 c n + 1 − 1 ( n + 2 ) ( n + 1 ) c n . Hence, y = ( c 0 + c 1 x + ( − c 1 − 1 2 c 0 ) x 2 + ( 1 2 c 1 + 1 3 c 0 ) x 3 + ( − 1 6 c 1 − 1 8 c 0 ) x 4 + ( 1 24 c 1 + 1 30 c 0 ) x 5 + ( − 1 120 c 1 − 1 144 c 0 ) x 6 + ⋯ ) = ( c 0 − 1 2 c 0 x 2 + 1 3 c 0 x 3 − 1 8 c 0 x 4 + 1 30 x 5 − 1 144 c 0 x 6 + ⋯ + c 1 x − c 1 x 2 + 1 2 c 1 x 3 − 1 6 c 1 x 3 − 1 6 c 1 x 4 + 1 24 c 1 x 5 − 1 120 x 6 + ⋯ ) = ( c 0 + <

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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 17.5, Problem 2E

To determine

To find: The general solution of the differential equation y″+2y′+y=0 by using power series.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.5, Problem 1E

Chapter 17.5, Problem 3E

Chapter 17 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 17.1 - In Exercises 1−30, find the general solution of...Ch. 17.1 - In Exercises 1 – 30, find the general solution of...Ch. 17.1 - In Exercises 1 – 30, find the general solution of...Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Prob. 9E Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - In Exercises 1 – 30, find the general solution of...Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Prob. 35E Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Prob. 38E Ch. 17.1 - Prob. 39E Ch. 17.1 - Prob. 40E Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Prob. 43E Ch. 17.1 - Prob. 44E Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - In Exercises 41−55, find the general solution. 47....Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.1 - Prob. 52E Ch. 17.1 - Prob. 53E Ch. 17.1 - Prob. 54E Ch. 17.1 - Prob. 55E Ch. 17.1 - In Exercises 56−60, solve the initial value...Ch. 17.1 - Prob. 57E Ch. 17.1 - Prob. 58E Ch. 17.1 - Prob. 59E Ch. 17.1 - Prob. 60E Ch. 17.1 - Prob. 61E Ch. 17.1 - Prob. 62E Ch. 17.1 - Prob. 63E Ch. 17.1 - Prob. 64E Ch. 17.1 - Prob. 65E Ch. 17.1 - Show that if a, b, and c are positive constants,...Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.2 - Prob. 29E Ch. 17.2 - Prob. 30E Ch. 17.2 - Prob. 31E Ch. 17.2 - Prob. 32E Ch. 17.2 - Prob. 33E Ch. 17.2 - Prob. 34E Ch. 17.2 - Prob. 35E Ch. 17.2 - Prob. 36E Ch. 17.2 - Prob. 37E Ch. 17.2 - Prob. 38E Ch. 17.2 - Prob. 39E Ch. 17.2 - Prob. 40E Ch. 17.2 - Prob. 41E Ch. 17.2 - Prob. 42E Ch. 17.2 - Prob. 43E Ch. 17.2 - Prob. 44E Ch. 17.2 - Prob. 45E Ch. 17.2 - Prob. 46E Ch. 17.2 - Prob. 47E Ch. 17.2 - Prob. 48E Ch. 17.2 - Prob. 49E Ch. 17.2 - Prob. 50E Ch. 17.2 - Prob. 51E Ch. 17.2 - Prob. 52E Ch. 17.2 - Prob. 53E Ch. 17.2 - Prob. 54E Ch. 17.2 - Prob. 55E Ch. 17.2 - Prob. 56E Ch. 17.2 - Prob. 57E Ch. 17.2 - Prob. 58E Ch. 17.2 - Prob. 59E Ch. 17.2 - Prob. 60E Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.4 - In Exercises 1–24, find the general solution to...Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Prob. 15E Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - Prob. 21E Ch. 17.4 - Prob. 22E Ch. 17.4 - In Exercises 1–24, find the general solution to...Ch. 17.4 - Prob. 24E Ch. 17.4 - Prob. 25E Ch. 17.4 - Prob. 26E Ch. 17.4 - Prob. 27E Ch. 17.4 - Prob. 28E Ch. 17.4 - Prob. 29E Ch. 17.4 - Prob. 30E Ch. 17.5 - Prob. 1E Ch. 17.5 - Prob. 2E Ch. 17.5 - Prob. 3E Ch. 17.5 - Prob. 4E Ch. 17.5 - Prob. 5E Ch. 17.5 - Prob. 6E Ch. 17.5 - Prob. 7E Ch. 17.5 - Prob. 8E Ch. 17.5 - Prob. 9E Ch. 17.5 - Prob. 10E Ch. 17.5 - Prob. 11E Ch. 17.5 - Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Prob. 17E Ch. 17.5 - Prob. 18E

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.

Recommended textbooks for you

Calculus: Early Transcendentals
Calculus
ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Precalculus
Calculus
ISBN:9780135189405
Author:Michael Sullivan
Publisher:PEARSON
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning

Calculus: Early Transcendentals

Calculus

ISBN:9781285741550

Author:James Stewart

Publisher:Cengage Learning

Thomas' Calculus (14th Edition)

Calculus

ISBN:9780134438986

Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher:PEARSON

Calculus: Early Transcendentals (3rd Edition)

Calculus

ISBN:9780134763644

Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher:PEARSON

Calculus: Early Transcendentals

Calculus

ISBN:9781319050740

Author:Jon Rogawski, Colin Adams, Robert Franzosa

Publisher:W. H. Freeman

Precalculus

Calculus

ISBN:9780135189405

Author:Michael Sullivan

Publisher:PEARSON

Calculus: Early Transcendental Functions

Calculus

ISBN:9781337552516

Author:Ron Larson, Bruce H. Edwards

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY

Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY

Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY

The general solution of the differential equation y ″ + 2 y ′ + y = 0 by using power series.

Concept explainers

Videos

To find: The general solution of the differential equation y″+2y′+y=0 by using power series.

Want to see the full answer?

Chapter 17 Solutions