Explanation Formula used: The auxiliary equation of a linear homogeneous differential equation of the form a 0 D 2 y + a 1 D y + a 2 y = 0 with constant coefficients is given by a 0 m 2 + a 1 m + a 2 = 0 . The general solution of the auxiliary equation with complex roots is given by y = e α x ( c 1 sin β x + c 2 cos β x ) . The general solution of the auxiliary equation with double roots m is given by y = e m x ( c 1 + c 2 x ) . The general solution of the auxiliary equation with two distinct roots m 1 and m 2 , is given by y = c 1 e m 1 x + c 2 e m 2 x . The solution of a linear non-homogeneous differential equation of the form a 0 D 2 y + a 1 D y + a 2 y = b where the a ’s are constants and b is a function of x or is a constant, is given by y = y c + y p . The y c is called the complementary solution obtained by solving the corresponding homogeneous equation and where y p is the particular solution. The different forms of b and their corresponding particular solution to be used is shown below: b y p ( x ) a e β x A e β x a cos ( β x ) A cos ( β x ) + B sin ( β x ) b sin ( β x ) A cos ( β x ) + B sin ( β x ) a cos ( β x ) + b sin ( β x ) A cos ( β x ) + B sin ( β x ) n th degree polynomial A n x n + A n − 1 x n − 1 + A 1 x + A 0 Calculation: The given differential equation is 6 D 2 y + D y − y = e x − e − x . Compare the given equation with a 0 D 2 y + a 1 D y + a 2 y = b and obtain the values of a 0 , a 1 , a 2 , b as shown below: a 0 = 6 , a 1 = 1 , a 2 = − 1 , b = e x − e − x . Write the auxiliary equation of the homogeneous equation 6 D 2 y + D y − y = 0 as, 6 m 2 + m − 1 = 0 Solve the auxiliary equation and obtain the roots as follows: 6 m 2 + m − 1 = 0 ( 2 m + 1 ) ( 3 m − 1 ) = 0 m 1 = − 1 2 , m 2 = 1 3 Thus, the auxiliary equation has two distinct roots m 1 = − 1 2 , m 2 = 1 3 . Using the above formula, obtain the complementary solution of the homogeneous equation 6 D 2 y + D y − y = 0 as, y c = c 1 e − x 2 + c 2 e x 3 Let y p = A e x + B e − x

11th Edition

ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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 31.9, Problem 22E

To determine

Solution the differential equation 6D2y+Dy−y=ex−e−x.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 31.9, Problem 21E

Chapter 31.9, Problem 23E

Students have asked these similar questions

Use mathematical induction to prove the following statement: For all natural numbers n, 5 divides 6^n - 1 (show every step in detail)

Solve this

Use mathematical induction to prove the following statement: For all natural numbers n, 5 divides 6^n - 1

Chapter 31 Solutions

EBK BASIC TECHNICAL MATHEMATICS

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

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.

Solution the differential equation 6 D 2 y + D y − y = e x − e − x .

Concept explainers

Videos

Solution the differential equation 6D2y+Dy−y=ex−e−x.

Want to see the full answer?

Chapter 31 Solutions