Explanation The given differential equation is y ″ + 2 y ′ + 5 y = 4 e − t cos 2 t . Consider the homogeneous equation y ″ + 2 y ′ + 5 y = 0 . Obtain the characteristic equation by replacing y ″ with r 2 , y ′ with r and y with 1 in the differential equation: r 2 + 2 r + 5 = 0 . The roots of the characteristic equation are as follows. r 2 + 2 r + 5 = 0 r 1 , 2 = − 2 ± ( 2 ) 2 − 4 ( 1 ) ( 5 ) 2 ( 1 ) ( ∵ x 1 , 2 = − b ± b 2 − 4 a c 2 a ) r 1 , 2 = − 2 ± 4 − 20 2 r 1 , 2 = − 2 ± − 16 2 Simplify further, r 1 , 2 = − 2 ± i 4 2 r 1 , 2 = − 1 ± 2 i ( ∵ r = a + i b ) Here, roots are complex. Note that, the general equation is y c ( t ) = c 1 e a t cos b t + c 2 e a t sin b t where a is the real part of the root and b is the imaginary part of the roots. Substitute a = − 1 and b = 2 in y c ( t ) = c 1 e a t cos b t + c 2 e a t sin b t . y c ( t ) = c 1 e − t cos ( 2 t ) + c 2 e − t sin ( 2 t ) Therefore, the complementary solution of the given differential equation is y c ( t ) = c 1 e − t cos ( 2 t ) + c 2 e − t sin ( 2 t ) . Let Y = A t e − t cos 2 t + B t e − t sin 2 t ( ∵ g ( t ) = 4 e − t cos 2 t ) . Obtain the first derivative of Y = A t e − t cos 2 t + B t e − t sin 2 t as follows. Y ′ = d d t ( A t e − t cos 2 t + B t e − t sin 2 t ) = e − t ( A cos 2 t + B sin 2 t ) − t e − t ( A cos 2 t + B sin 2 t ) + t e − t ( − 2 A sin 2 t + 2 B cos 2 t ) = e − t ( A cos 2 t + B sin 2 t ) + t e − t ( ( − A + 2 B ) cos 2 t + ( − 2 A − B ) sin 2 t ) Obtain the second derivative of Y = A t e − t cos 2 t + B t e − t sin 2 t as follows. Y ′ ′ = d d t ( Y ′ ) = d d t ( e − t ( A cos 2 t + B sin 2 t ) + t e − t ( ( − A + 2 B ) cos 2 t + ( − 2 A − B ) sin 2 t ) ) = [ − e − t ( A cos 2 t + B sin 2 t ) + e − t ( − 2 A sin 2 t + 2 B cos 2 t ) + e − t ( ( − A + 2 B ) cos 2 t + ( − 2 A − B ) sin 2 t ) − t e − t ( ( − A + 2 B ) cos 2 t + ( − 2 A − B ) sin 2 t ) + t e − t ( − 2 ( − A + 2 B ) sin 2 t + 2 ( − 2 A − B ) cos 2 t ) ] = [ e − t cos 2 t ( − A + 2 B − A + 2 B ) + e − t sin 2 t ( − B − 2 A − 2 A − B ) + t e − t cos 2 t ( A − 2 B − 4 A − 2 B ) + t e − t sin 2 t ( 2 A + B + 2 A − 4 B ) ] = [ e − t cos 2 t ( − 2 A + 4 B ) + e − t sin 2 t ( − 2 B − 4 A ) + t e − t cos 2 t ( − 3 A − 4 B ) + t e − t sin 2 t ( 4 A − 3 B ) ] Obtain the particular solution as follows. Substitute Y ′ = e − t ( A cos 2 t + B sin 2 t ) + t e − t ( ( − A + 2 B ) cos 2 t + ( − 2 A − B ) sin 2 t ) , Y ′ ′ = [ e − t cos 2 t ( − 2 A + 4 B ) + e − t sin 2 t ( − 2 B − 4 A ) + t e − t cos 2 t ( − 3 A − 4 B ) + t e − t sin 2 t ( 4 A − 3 B ) ] and Y = A t e − t cos 2 t + B t e − t sin 2 t in y ″ + 2 y ′ + 5 y = 4 e − t cos 2 t

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 3 Solutions

Elementary Differential Equations and Boundary Value Problems, Enhanced

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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