Explanation The given differential equation is y ″ − 3 y ′ − 4 y = 3 e 2 t . Consider the homogeneous equation y ″ − 3 y ′ − 4 y = 0 . Obtain the characteristic equation by replacing y ″ with r 2 , y ′ with r and y with 1 in the differential equation as r 2 − 3 r − 4 = 0 . The roots of the characteristic equation are as follows. r 2 − 3 r − 4 = 0 r 1 , 2 = 3 ± ( − 3 ) 2 − 4 ( 1 ) ( − 4 ) 2 ( 1 ) ( ∵ x 1 , 2 = − b ± b 2 − 4 a c 2 a ) r 1 , 2 = 3 ± 9 + 16 2 r 1 , 2 = 3 ± 25 2 Simplify further as follows. r 1 , 2 = 3 + 5 2 , 3 − 5 2 r 1 , 2 = 4 , − 1 r 1 = − 1 or r 2 = 4 Note that, the general equation is y c ( t ) = c 1 e r 1 t + c 2 e r 2 t where r 1 and r 2 are the distinct root of the characteristic equation. Substitute r 1 = − 1 and r 2 = 4 in y c ( t ) = c 1 e r 1 t + c 2 e r 2 t . y c ( t ) = c 1 e − t + c 2 e 4 t Thus, the complementary solution is y c ( t ) = c 1 e − t + c 2 e 4 t . Compare y ″ − 3 y ′ − 4 y = 3 e 2 t with standard form y ″ + b y ′ + c y = g ( t ) . Here, b = − 3 , c = − 4 and g ( t ) = 3 e 2 t . Substitute b = − 3 , c = − 4 in r 1 + r 2 = − b , r 1 r 2 = c . r 1 + r 2 = 3 r 1 r 2 = − 4 Thus, the value of r 1 and r 2 is − 1 and 4 . Obtain the solution of the equation ( D − r 1 ) u = g ( t ) as follows. Substitute r 1 = − 1 and g ( t ) = 3 e 2 t in ( D − r 1 ) u = g ( t ) . ( D + 1 ) u = 3 e 2 t d u d t + u = 3 e 2 t Note that, the general equation is y ′ + p ( x ) y = q ( x ) , and the integral factor is μ = e ∫ p ( x ) d x , then the general solution is y μ = ∫ q ( x ) μ d x

Elementary Differential Equations And Boundary Value Problems

11th Edition

ISBN: 9781119256007

Author: Boyce, William E.

Publisher: John Wiley & Sons, Incorporated

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Question

Chapter 3.5, Problem 28P

To determine

The general solution of the given differential equation.

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Chapter 3.5, Problem 27P

Chapter 3.5, Problem 29P

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Chapter 3 Solutions

Elementary Differential Equations And Boundary Value Problems

Ch. 3.1 - Prob. 1P Ch. 3.1 - Prob. 2P Ch. 3.1 - Prob. 3P Ch. 3.1 - Prob. 4P Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 7P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P

Ch. 3.1 - Prob. 11P Ch. 3.1 - Prob. 12P Ch. 3.1 - Prob. 13P Ch. 3.1 - Prob. 14P Ch. 3.1 - Prob. 15P Ch. 3.1 - Prob. 16P Ch. 3.1 - Prob. 17P Ch. 3.1 - Prob. 18P Ch. 3.1 - Prob. 20P Ch. 3.2 - Prob. 1P Ch. 3.2 - Prob. 2P Ch. 3.2 - Prob. 3P Ch. 3.2 - Prob. 4P Ch. 3.2 - Prob. 5P Ch. 3.2 - Prob. 6P Ch. 3.2 - Prob. 7P Ch. 3.2 - Prob. 8P Ch. 3.2 - Prob. 9P Ch. 3.2 - Prob. 10P Ch. 3.2 - Prob. 11P Ch. 3.2 - Prob. 12P Ch. 3.2 - Prob. 13P Ch. 3.2 - Prob. 14P Ch. 3.2 - Prob. 15P Ch. 3.2 - Prob. 16P Ch. 3.2 - Prob. 17P Ch. 3.2 - Prob. 18P Ch. 3.2 - Prob. 19P Ch. 3.2 - Prob. 20P Ch. 3.2 - Prob. 21P Ch. 3.2 - Prob. 22P Ch. 3.2 - Prob. 23P Ch. 3.2 - Prob. 24P Ch. 3.2 - Prob. 25P Ch. 3.2 - Prob. 26P Ch. 3.2 - Prob. 27P Ch. 3.2 - Prob. 28P Ch. 3.2 - Prob. 29P Ch. 3.2 - Prob. 30P Ch. 3.2 - Prob. 31P Ch. 3.2 - Prob. 32P Ch. 3.2 - Prob. 33P Ch. 3.2 - Prob. 34P Ch. 3.2 - Prob. 35P Ch. 3.2 - Prob. 36P Ch. 3.2 - Prob. 37P Ch. 3.2 - Prob. 38P Ch. 3.3 - Prob. 1P Ch. 3.3 - Prob. 2P Ch. 3.3 - Prob. 3P Ch. 3.3 - Prob. 4P Ch. 3.3 - Prob. 5P Ch. 3.3 - Prob. 6P Ch. 3.3 - Prob. 7P Ch. 3.3 - Prob. 8P Ch. 3.3 - Prob. 9P Ch. 3.3 - Prob. 10P Ch. 3.3 - Prob. 11P Ch. 3.3 - Prob. 12P Ch. 3.3 - Prob. 13P Ch. 3.3 - Prob. 14P Ch. 3.3 - Prob. 15P Ch. 3.3 - Prob. 16P Ch. 3.3 - Prob. 17P Ch. 3.3 - Prob. 18P Ch. 3.3 - Prob. 19P Ch. 3.3 - Prob. 20P Ch. 3.3 - Prob. 21P Ch. 3.3 - Prob. 22P Ch. 3.3 - Prob. 23P Ch. 3.3 - Prob. 24P Ch. 3.3 - Prob. 25P Ch. 3.3 - Prob. 26P Ch. 3.3 - Prob. 27P Ch. 3.3 - Prob. 28P Ch. 3.3 - Prob. 29P Ch. 3.3 - Prob. 30P Ch. 3.3 - Prob. 31P Ch. 3.3 - Prob. 32P Ch. 3.3 - Prob. 33P Ch. 3.3 - Prob. 34P Ch. 3.3 - Prob. 35P Ch. 3.3 - Prob. 36P Ch. 3.4 - Prob. 1P Ch. 3.4 - Prob. 2P Ch. 3.4 - Prob. 3P Ch. 3.4 - Prob. 4P Ch. 3.4 - Prob. 5P Ch. 3.4 - Prob. 6P Ch. 3.4 - Prob. 7P Ch. 3.4 - Prob. 8P Ch. 3.4 - Prob. 9P Ch. 3.4 - Prob. 10P Ch. 3.4 - Prob. 11P Ch. 3.4 - Prob. 12P Ch. 3.4 - Prob. 13P Ch. 3.4 - Prob. 14P Ch. 3.4 - Prob. 15P Ch. 3.4 - Prob. 16P Ch. 3.4 - Prob. 17P Ch. 3.4 - Prob. 18P Ch. 3.4 - Prob. 19P Ch. 3.4 - Prob. 20P Ch. 3.4 - Prob. 21P Ch. 3.4 - Prob. 22P Ch. 3.4 - Prob. 23P Ch. 3.4 - Prob. 24P Ch. 3.4 - Prob. 25P Ch. 3.4 - Prob. 26P Ch. 3.4 - Prob. 27P Ch. 3.4 - Prob. 28P Ch. 3.4 - Prob. 29P Ch. 3.4 - Prob. 30P Ch. 3.4 - Prob. 31P Ch. 3.4 - Prob. 32P Ch. 3.4 - Prob. 33P Ch. 3.4 - Prob. 34P Ch. 3.5 - Prob. 1P Ch. 3.5 - Prob. 2P Ch. 3.5 - Prob. 3P Ch. 3.5 - Prob. 4P Ch. 3.5 - Prob. 5P Ch. 3.5 - Prob. 6P Ch. 3.5 - Prob. 7P Ch. 3.5 - Prob. 8P Ch. 3.5 - Prob. 9P Ch. 3.5 - Prob. 10P Ch. 3.5 - Prob. 11P Ch. 3.5 - Prob. 12P Ch. 3.5 - Prob. 13P Ch. 3.5 - Prob. 14P Ch. 3.5 - Prob. 15P Ch. 3.5 - Prob. 22P Ch. 3.5 - Prob. 23P Ch. 3.5 - Prob. 24P Ch. 3.5 - Prob. 25P Ch. 3.5 - Prob. 26P Ch. 3.5 - Prob. 27P Ch. 3.5 - Prob. 28P Ch. 3.5 - Prob. 29P Ch. 3.5 - Prob. 30P Ch. 3.6 - Prob. 1P Ch. 3.6 - Prob. 2P Ch. 3.6 - Prob. 3P Ch. 3.6 - Prob. 4P Ch. 3.6 - Prob. 5P Ch. 3.6 - Prob. 6P Ch. 3.6 - Prob. 7P Ch. 3.6 - Prob. 8P Ch. 3.6 - Prob. 9P Ch. 3.6 - Prob. 10P Ch. 3.6 - Prob. 11P Ch. 3.6 - Prob. 12P Ch. 3.6 - Prob. 13P Ch. 3.6 - Prob. 14P Ch. 3.6 - Prob. 15P Ch. 3.6 - Prob. 16P Ch. 3.6 - Prob. 17P Ch. 3.6 - Prob. 18P Ch. 3.6 - Prob. 19P Ch. 3.6 - Prob. 20P Ch. 3.6 - Prob. 21P Ch. 3.6 - Prob. 22P Ch. 3.6 - Prob. 23P Ch. 3.6 - Prob. 24P Ch. 3.6 - Prob. 25P Ch. 3.6 - Prob. 26P Ch. 3.7 - Prob. 1P Ch. 3.7 - Prob. 2P Ch. 3.7 - Prob. 3P Ch. 3.7 - Prob. 4P Ch. 3.7 - Prob. 5P Ch. 3.7 - Prob. 6P Ch. 3.7 - Prob. 7P Ch. 3.7 - Prob. 8P Ch. 3.7 - Prob. 9P Ch. 3.7 - Prob. 10P Ch. 3.7 - Prob. 11P Ch. 3.7 - Prob. 12P Ch. 3.7 - Prob. 13P Ch. 3.7 - Prob. 14P Ch. 3.7 - Prob. 15P Ch. 3.7 - Prob. 16P Ch. 3.7 - Prob. 17P Ch. 3.7 - Prob. 18P Ch. 3.7 - Prob. 19P Ch. 3.7 - Prob. 20P Ch. 3.7 - Prob. 21P Ch. 3.7 - Prob. 22P Ch. 3.7 - Prob. 23P Ch. 3.7 - Prob. 24P Ch. 3.8 - Prob. 1P Ch. 3.8 - Prob. 2P Ch. 3.8 - Prob. 3P Ch. 3.8 - Prob. 4P Ch. 3.8 - Prob. 5P Ch. 3.8 - Prob. 6P Ch. 3.8 - Prob. 7P Ch. 3.8 - Prob. 9P Ch. 3.8 - Prob. 11P Ch. 3.8 - Prob. 14P

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