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 D 2 y − 2 D y − 8 y = 4 e − 2 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 = 1 , a 1 = − 2 , a 2 = − 8 , b = 4 e − 2 x . Write the auxiliary equation of the homogeneous equation D 2 y − 2 D y − 8 y = 0 as, m 2 − 2 m − 8 = 0 Solve the auxiliary equation and obtain the roots as follows: m 2 − 2 m − 8 = 0 ( m + 2 ) ( m − 4 ) = 0 m 1 = − 2 , m 2 = 4 Thus, the auxiliary equation has distinct roots m 1 = − 2 , and m 2 = 4

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 31, Problem 6PT

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To solve: The differential equation D2y−2Dy−8y=4e−2x.

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Chapter 31 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

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 =...

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The differential equation D 2 y − 2 D y − 8 y = 4 e − 2 x .

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Chapter 31 Solutions