Program Explanation Given information: The volume of tank 1 is 60 gal , volume of tank 2 is 20 gal and volume of tank 3 is 30 gal . The value of r is 60 gal / min , x 1 ( 0 ) = 55 , x 2 ( 0 ) = 0 and x 3 ( 0 ) = 0 . Explanation: Consider the rate equation for the tank 1 as follows. d x 1 d t = − r V 1 x 1 + r V 3 x 3 ......... (1) Consider the rate equation for the tank 2 as follows. d x 2 d t = r V 1 x 1 − r V 2 x 2 ......... (2) Consider the rate equation for the tank 3 as follows. d x 3 d t = r V 2 x 2 − r V 3 x 3 ......... (3) Substitute 60 for r, 60 gal for V 1 and 30 gal for V 3 in equation (1). d x 1 d t = − 60 60 x 1 + 60 30 x 3 = − x 1 + 2 x 3 Substitute 60 for r , 20 gal for V 2 and 60 gal for V 1 in equation (2). d x 2 d t = 60 60 x 1 − 60 20 x 2 = x 1 − 3 x 2 Substitute 10 for r , 20 gal for V 2 and 30 gal for V 3 in equation (3). d x 3 d t = 60 20 x 2 − 60 30 x 3 = 3 x 2 − 2 x 3 Consider matrix form of the system of equation as A. A x = [ − 1 0 2 1 − 3 0 0 3 − 2 ] x The characteristics equation of the coefficient matrix is, [ − 1 − λ 0 2 1 − 3 − λ 0 0 3 − 2 − λ ] = 0 ( − 1 − λ ) ( − 3 − λ ) ( − 2 − λ ) + 2 ( 3 − 0 ) = 0 − ( λ+ 1 ) ( λ + 3 ) ( λ+ 2 ) + 6 = 0 λ = − 3 + 2 i or λ = − 3 − 2 i or λ = 0 The Eigen vector of the coefficient matrix A is in the form as, [ − 1 − λ 0 2 1 − 3 − λ 0 0 3 − 2 − λ ] [ a b c ] = [ 0 0 0 ] The associated Eigen vector with the matrix Ais v = [ a b c ] T . Substitute 0 for λ in coefficient matrix, [ − 1 − 0 0 2 1 − 3 − 0 0 0 3 − 2 − 0 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ − 1 0 2 1 − 3 0 0 3 − 2 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, a 1 − 2 c 1 = 0 a 1 − 3 b 1 = 0 3 b 1 − 2 c 1 = 0 After solving the equation the value of a 1 , b 1 and c 1 can be obtained as, a 1 = 6 b 1 = 2 c 1 = 3 Therefore, the associated Eigen vector is shown below. v 1 = [ 6 2 3 ] Substitute − 3 + 2 i for λ in coefficient matrix, [ − 1 − ( − 3 + 2 i ) 0 2 1 − 3 − ( − 3 + 2 i ) 0 0 3 − 2 − ( − 3 + 2 i ) ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] [ 2 + 2 i 0 2 1 − 2 i 0 0 3 1 − 2 i ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] The scalar equation are shown below, ( 2 − 2 i ) a 2 + 2 c 2 = 0 a 2 − ( 2 i ) b 2 = 0 3 b 2 + ( 1 − 2 i ) c 2 = 0 After solving the equation the value of a 2 , b 2 and c 2 can be obtained as, a 2 = 2 cos 2 t − 2 sin 2 t b 2 = cos 2 t + 2 sin 2 t c 2 = − 3 cos 2 t Therefore, the associated Eigen vector is shown below. v 2 = [ 2 cos 2 t − 2 sin 2 t cos 2 t + 2 sin 2 t − 3 cos 2 t ] Substitute − 3 − 2 i for λ in coefficient matrix, [ − 1 − ( − 3 − 2 i ) 0 2 1 − 3 − ( − 3 − 2 i ) 0 0 3 − 2 − ( − 3 − 2 i ) ] [ a 3 b 3 c 3 ] = [ 0 0 0 ] [ ( 2 + 2 i ) 0 2 1 ( 2 i ) 0 0 3 ( 1 + 2 i ) ] [ a 3 b 3 c 3 ] = [ 0 0 0 ] The scalar equation are shown below, ( 2 − 2 i ) a 3 + 2 c 3 = 0 a 3 + ( 2 i ) b 3 = 0 3 b 3 + ( 1 + 2 i ) c 3 = 0 After solving the equation the value of a 3 , b 3 and c 3 can be obtained as, a 3 = 2 sin 2 t − 2 cos 2 t b 3 = sin 2 t − 2 cos 2 t c 3 = − 3 sin 2 t Therefore, the associated Eigen vector is shown below. v 3 = [ 2 sin 2 t + 2 cos 2 t sin 2 t − 2 cos 2 t − 3 sin 2 t ] The three solution of the equation can be obtained as, [ x 1 ( t ) = [ 6 2 3 ] e ( 0 ) t , x 2 ( t ) = [ 2 cos 2 t − 2 sin 2 t cos 2 t + 2 sin 2 t − 3 cos 2 t ] e ( − 3 ) t and x 3 ( t ) = [ 2 sin 2 t + 2 cos 2 t sin 2 t − 2 cos 2 t − 3 sin 2 t ] e ( − 3 ) t ] The general solution of the system equation is defined as, x ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) + c 3 x 3 ( t ) Substitute the value of x 1 ( t ) , x 2 ( t ) and x 3 ( t ) in above equation

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 5.2, Problem 37P

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Program Description:Purpose of problem is to obtain the amount of salt in three tank at time t , the limiting amount of salt in each water tank and also draw a graph of x1(t),x2(t) and x3(t) .

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Chapter 5.2, Problem 36P

Chapter 5.2, Problem 38P

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Chapter 5 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

Ch. 5.1 - Prob. 11P Ch. 5.1 - Prob. 12P Ch. 5.1 - Prob. 13P Ch. 5.1 - Prob. 14P Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - Prob. 17P Ch. 5.1 - Prob. 18P Ch. 5.1 - Prob. 19P Ch. 5.1 - Prob. 20P Ch. 5.1 - Prob. 21P Ch. 5.1 - Prob. 22P Ch. 5.1 - Prob. 23P Ch. 5.1 - Prob. 24P Ch. 5.1 - Prob. 25P Ch. 5.1 - Prob. 26P Ch. 5.1 - Prob. 27P Ch. 5.1 - Prob. 28P Ch. 5.1 - Prob. 29P Ch. 5.1 - Prob. 30P Ch. 5.1 - Prob. 31P Ch. 5.1 - Prob. 32P Ch. 5.1 - Prob. 33P Ch. 5.1 - Prob. 34P Ch. 5.1 - Prob. 35P Ch. 5.1 - Prob. 36P Ch. 5.1 - Prob. 37P Ch. 5.1 - Prob. 38P Ch. 5.1 - Prob. 39P Ch. 5.1 - Prob. 40P Ch. 5.1 - Prob. 41P Ch. 5.1 - Prob. 42P Ch. 5.1 - Prob. 43P Ch. 5.1 - Prob. 44P Ch. 5.1 - Prob. 45P Ch. 5.2 - Prob. 1P Ch. 5.2 - Prob. 2P Ch. 5.2 - Prob. 3P Ch. 5.2 - Prob. 4P Ch. 5.2 - Prob. 5P Ch. 5.2 - Prob. 6P Ch. 5.2 - Prob. 7P Ch. 5.2 - Prob. 8P Ch. 5.2 - Prob. 9P Ch. 5.2 - Prob. 10P Ch. 5.2 - Prob. 11P Ch. 5.2 - Prob. 12P Ch. 5.2 - Prob. 13P Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.2 - Prob. 16P Ch. 5.2 - Prob. 17P Ch. 5.2 - Prob. 18P Ch. 5.2 - Prob. 19P Ch. 5.2 - Prob. 20P Ch. 5.2 - Prob. 21P Ch. 5.2 - Prob. 22P Ch. 5.2 - Prob. 23P Ch. 5.2 - Prob. 24P Ch. 5.2 - Prob. 25P Ch. 5.2 - Prob. 26P Ch. 5.2 - Prob. 27P Ch. 5.2 - Prob. 28P Ch. 5.2 - Prob. 29P Ch. 5.2 - Prob. 30P Ch. 5.2 - Prob. 31P Ch. 5.2 - Prob. 32P Ch. 5.2 - Prob. 33P Ch. 5.2 - Prob. 34P Ch. 5.2 - Prob. 35P Ch. 5.2 - Prob. 36P Ch. 5.2 - Prob. 37P Ch. 5.2 - Prob. 38P Ch. 5.2 - Prob. 39P Ch. 5.2 - Prob. 40P Ch. 5.2 - Prob. 41P Ch. 5.2 - Prob. 42P Ch. 5.2 - Prob. 43P Ch. 5.2 - Prob. 44P Ch. 5.2 - Prob. 45P Ch. 5.2 - Prob. 46P Ch. 5.2 - Prob. 47P Ch. 5.2 - Prob. 48P Ch. 5.2 - Prob. 49P Ch. 5.2 - Prob. 50P Ch. 5.3 - Prob. 1P Ch. 5.3 - Prob. 2P Ch. 5.3 - Prob. 3P Ch. 5.3 - Prob. 4P Ch. 5.3 - Prob. 5P Ch. 5.3 - Prob. 6P Ch. 5.3 - Prob. 7P Ch. 5.3 - Prob. 8P Ch. 5.3 - Prob. 9P Ch. 5.3 - Prob. 10P Ch. 5.3 - Prob. 11P Ch. 5.3 - Prob. 12P Ch. 5.3 - Prob. 13P Ch. 5.3 - Prob. 14P Ch. 5.3 - Prob. 15P Ch. 5.3 - Prob. 16P Ch. 5.3 - Prob. 17P Ch. 5.3 - Prob. 18P Ch. 5.3 - Prob. 19P Ch. 5.3 - Prob. 20P Ch. 5.3 - Prob. 21P Ch. 5.3 - Prob. 22P Ch. 5.3 - Prob. 23P Ch. 5.3 - Prob. 24P Ch. 5.3 - Prob. 25P Ch. 5.3 - Prob. 26P Ch. 5.3 - Prob. 27P Ch. 5.3 - Prob. 28P Ch. 5.3 - Prob. 29P Ch. 5.3 - Prob. 30P Ch. 5.3 - Prob. 31P Ch. 5.3 - Prob. 32P Ch. 5.3 - Prob. 33P Ch. 5.3 - Verify Eq. (53) by substituting the expressions...Ch. 5.3 - Prob. 35P Ch. 5.3 - Prob. 36P Ch. 5.3 - Prob. 37P Ch. 5.3 - Prob. 38P Ch. 5.3 - Prob. 39P Ch. 5.3 - Prob. 40P Ch. 5.4 - Prob. 1P Ch. 5.4 - Prob. 2P Ch. 5.4 - Prob. 3P Ch. 5.4 - Prob. 4P Ch. 5.4 - Prob. 5P Ch. 5.4 - Prob. 6P Ch. 5.4 - Prob. 7P Ch. 5.4 - Prob. 8P Ch. 5.4 - Prob. 9P Ch. 5.4 - Prob. 10P Ch. 5.4 - Prob. 11P Ch. 5.4 - Prob. 12P Ch. 5.4 - Prob. 13P Ch. 5.4 - Prob. 14P Ch. 5.4 - Prob. 15P Ch. 5.4 - Prob. 16P Ch. 5.4 - Prob. 17P Ch. 5.4 - Prob. 18P Ch. 5.4 - Prob. 19P Ch. 5.4 - Prob. 20P Ch. 5.4 - Prob. 21P Ch. 5.4 - Prob. 22P Ch. 5.4 - Prob. 23P Ch. 5.4 - Prob. 24P Ch. 5.4 - Prob. 25P Ch. 5.4 - Prob. 26P Ch. 5.4 - Prob. 27P Ch. 5.4 - Prob. 28P Ch. 5.4 - Prob. 29P Ch. 5.5 - Prob. 1P Ch. 5.5 - Prob. 2P Ch. 5.5 - Prob. 3P Ch. 5.5 - Prob. 4P Ch. 5.5 - Prob. 5P Ch. 5.5 - Prob. 6P Ch. 5.5 - Prob. 7P Ch. 5.5 - Prob. 8P Ch. 5.5 - Prob. 9P Ch. 5.5 - Prob. 10P Ch. 5.5 - Prob. 11P Ch. 5.5 - Prob. 12P Ch. 5.5 - Prob. 13P Ch. 5.5 - Prob. 14P Ch. 5.5 - Prob. 15P Ch. 5.5 - Prob. 16P Ch. 5.5 - Prob. 17P Ch. 5.5 - Prob. 18P Ch. 5.5 - Prob. 19P Ch. 5.5 - Prob. 20P Ch. 5.5 - Prob. 21P Ch. 5.5 - Prob. 22P Ch. 5.5 - Prob. 23P Ch. 5.5 - Prob. 24P Ch. 5.5 - Prob. 25P Ch. 5.5 - Prob. 26P Ch. 5.5 - Prob. 27P Ch. 5.5 - Prob. 28P Ch. 5.5 - Prob. 29P Ch. 5.5 - Prob. 30P Ch. 5.5 - Prob. 31P Ch. 5.5 - Prob. 32P Ch. 5.5 - Prob. 33P Ch. 5.5 - Prob. 34P Ch. 5.5 - Prob. 35P Ch. 5.5 - Prob. 36P Ch. 5.6 - Prob. 1P Ch. 5.6 - Prob. 2P Ch. 5.6 - Prob. 3P Ch. 5.6 - Prob. 4P Ch. 5.6 - Prob. 5P Ch. 5.6 - Prob. 6P Ch. 5.6 - Prob. 7P Ch. 5.6 - Prob. 8P Ch. 5.6 - Prob. 9P Ch. 5.6 - Prob. 10P Ch. 5.6 - Prob. 11P Ch. 5.6 - Prob. 12P Ch. 5.6 - Prob. 13P Ch. 5.6 - Prob. 14P Ch. 5.6 - Prob. 15P Ch. 5.6 - Prob. 16P Ch. 5.6 - Prob. 17P Ch. 5.6 - Prob. 18P Ch. 5.6 - Prob. 19P Ch. 5.6 - Prob. 20P Ch. 5.6 - Prob. 21P Ch. 5.6 - Prob. 22P Ch. 5.6 - Prob. 23P Ch. 5.6 - Prob. 24P Ch. 5.6 - Prob. 25P Ch. 5.6 - Prob. 26P Ch. 5.6 - Prob. 27P Ch. 5.6 - Prob. 28P Ch. 5.6 - Prob. 29P Ch. 5.6 - Prob. 30P Ch. 5.6 - Prob. 31P Ch. 5.6 - Prob. 32P Ch. 5.6 - Prob. 33P Ch. 5.6 - Prob. 34P Ch. 5.6 - Prob. 35P Ch. 5.6 - Prob. 36P Ch. 5.6 - Prob. 37P Ch. 5.6 - Prob. 38P Ch. 5.6 - Prob. 39P Ch. 5.6 - Prob. 40P Ch. 5.7 - Prob. 1P Ch. 5.7 - Prob. 2P Ch. 5.7 - Prob. 3P Ch. 5.7 - Prob. 4P Ch. 5.7 - Prob. 5P Ch. 5.7 - Prob. 6P Ch. 5.7 - Prob. 7P Ch. 5.7 - Prob. 8P Ch. 5.7 - Prob. 9P Ch. 5.7 - Prob. 10P Ch. 5.7 - Prob. 11P Ch. 5.7 - Prob. 12P Ch. 5.7 - Prob. 13P Ch. 5.7 - Prob. 14P Ch. 5.7 - Prob. 15P Ch. 5.7 - Prob. 16P Ch. 5.7 - Prob. 17P Ch. 5.7 - Prob. 18P Ch. 5.7 - Prob. 19P Ch. 5.7 - Prob. 20P Ch. 5.7 - Prob. 21P Ch. 5.7 - Prob. 22P Ch. 5.7 - Prob. 23P Ch. 5.7 - Prob. 24P Ch. 5.7 - Prob. 25P Ch. 5.7 - Prob. 26P Ch. 5.7 - Prob. 27P Ch. 5.7 - Prob. 28P Ch. 5.7 - Prob. 29P Ch. 5.7 - Prob. 30P Ch. 5.7 - Prob. 31P Ch. 5.7 - Prob. 32P Ch. 5.7 - Prob. 33P Ch. 5.7 - Prob. 34P

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Program Description: Purpose of problem is to obtain the amount of salt in three tank at time t , the limiting amount of salt in each water tank and also draw a graph of x 1 ( t ) , x 2 ( t ) and x 3 ( t ) .

Solution Summary: The author explains that the problem is to obtain the amount of salt in three water tanks at time t and also draw a graph of x_1(t)and

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Program Description:Purpose of problem is to obtain the amount of salt in three tank at time t , the limiting amount of salt in each water tank and also draw a graph of x1(t),x2(t) and x3(t) .

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Chapter 5 Solutions