Explanation Given information: The given differential equation is y ' ' − y ' − 2 y = 1 + x e − x and the initial conditions are y ( 0 ) = 1 , y ' ( 0 ) = 3 . Concept Used: If y p is the particular solution of the differential equation y ' ' + a y ' + b y = F ( x ) and y h is the general solution, then y = y h + y p is the general solution of the non homogeneous equation. Calculation: The characteristics equation is m 2 − m − 2 = 0 . First, we have to find the solution of the characteristics equation m 2 − m − 2 = 0 . m 2 − m − 2 = 0 m 2 − 2 m + m − 2 = 0 m ( m − 2 ) + 1 ( m − 2 ) = 0 ( m − 2 ) ( m + 1 ) = 0 m = 2 or m = − 1 Therefore, y h = C 1 e 2 x + C 2 e − x Let y p be generalized form of 1 + x e − x . y p = A + B x e − x + C x 2 e − x y p ' = − B x e − x + B e − x + 2 x C e − x − C x 2 e − x y p ' ' = B x e − x − B e − x − B e − x − 2 x C e − x + 2 C e − x + C x 2 e − x − 2 x C e − x y p ' ' = B x e − x − 2 B e − x − 4 x C e − x + 2 C e − x + C x 2 e − x Substituting in the original equation yields y ' ' − y ' − 2 y = 1 + x e − x B x e − x − 2 B e − x − 4 x C e − x + 2 C e − x + C x 2 e − x + B x e − x − B e − x − 2 x C e − x + C x 2 e − x − 2 A − 2 B x e − x − 2 C x 2

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ISBN: 8220103600781

Author: Edwards

Publisher: YUZU

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Chapter 16, Problem 49RE

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To calculate: The solution of the given differential equation y''−y'−2y=1+xe−x by the method of undetermined coefficients subject to the initial conditions y(0)=1,y'(0)=3.

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Chapter 16, Problem 48RE

Chapter 16, Problem 50RE

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Chapter 16 Solutions

EBK 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

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The solution of the given differential equation y ' ' − y ' − 2 y = 1 + x e − x by the method of undetermined coefficients subject to the initial conditions y ( 0 ) = 1 , y ' ( 0 ) = 3 .

Solution Summary: The author explains how to calculate the solution of the given differential equation y' 's 2y=1+xe-x by the method of undetermined coefficients.

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To calculate: The solution of the given differential equation y''−y'−2y=1+xe−x by the method of undetermined coefficients subject to the initial conditions y(0)=1,y'(0)=3.

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Chapter 16 Solutions