Program Explanation Given information: The volume of tank 1 is 20 gal , volume of tank 2 is 6 gal and volume of tank 3 is 40 gal . The value of r is 120 gal / min , x 1 ( 0 ) = 33 , 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 120 for r, 20 gal for V 1 and 40 gal for V 3 in equation (1). d x 1 d t = − 120 20 x 1 + 120 40 x 3 = − 6 x 1 + 3 x 3 Substitute 120 for r , 6 gal for V 2 and 20 gal for V 1 in equation (2). d x 2 d t = 120 20 x 1 − 120 6 x 2 = 6 x 1 − 20 x 2 Substitute 120 for r , 6 gal for V 2 and 40 gal for V 3 in equation (3). d x 3 d t = 120 6 x 2 − 120 40 x 3 = 20 x 2 − 3 x 3 Consider matrix form of the system of equation as A. A x = [ − 6 0 3 6 − 20 0 0 20 − 3 ] x The characteristics equation of the coefficient matrix is, [ − 6 − λ 0 3 6 − 20 − λ 0 0 20 − 3 − λ ] = 0 ( − 6 − λ ) ( − 20 − λ ) ( − 3 − λ ) + 360 = 0 ( λ+11 ) ( λ + 18 ) ( λ ) = 0 λ = − 18 or λ = − 11 or λ = 0 The Eigen vector of the coefficient matrix A is in the form as, [ − 6 − λ 0 3 6 − 20 − λ 0 0 20 − 3 − λ ] [ a b c ] = [ 0 0 0 ] The associated Eigen vector with the matrix Ais v = [ a b c ] T . Substitute − 18 for λ in coefficient matrix, [ − 6 + 18 0 3 6 − 20 + 18 0 0 20 − 3 + 18 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ 12 0 3 6 − 2 0 0 20 15 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, 12 a 1 − 3 c 1 = 0 6 a 1 − 2 b 1 = 0 20 b 1 + 15 c 1 = 0 After solving the equation the value of a 1 , b 1 and c 1 can be obtained as, a 1 = 1 b 1 = 3 c 1 = − 4 Therefore, the associated Eigen vector is shown below. v 1 = [ 1 3 − 4 ] Substitute − 11 for λ in coefficient matrix, [ − 6 + 11 0 3 6 − 20 + 11 0 0 20 − 3 + 11 ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] [ 5 0 3 6 − 9 0 0 20 8 ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] The scalar equation are shown below, 5 a 2 + 3 c 2 = 0 6 a 2 − 9 b 2 = 0 20 b 2 + 8 c 2 = 0 After solving the equation the value of a 2 , b 2 and c 2 can be obtained as, a 2 = 3 b 2 = 2 c 2 = − 5 Therefore, the associated Eigen vector is shown below. v 2 = [ 3 2 − 5 ] Substitute 0 for λ in coefficient matrix, [ − 6 − 0 0 3 6 − 20 − 0 0 0 20 − 3 − 0 ] [ a 3 b 3 c 3 ] = [ 0 0 0 ] [ − 6 0 3 6 − 20 0 0 20 − 3 ] [ a 3 b 3 c 3 ] = [ 0 0 0 ] The scalar equation are shown below, − 6 a 3 + 3 c 3 = 0 6 a 3 − 20 b 3 = 0 20 b 3 − 3 c 3 = 0 After solving the equation the value of a 3 , b 3 and c 3 can be obtained as, a 3 = 10 b 3 = 3 c 3 = 20 Therefore, the associated Eigen vector is shown below. v 3 = [ 10 3 20 ] The three solution of the equation can be obtained as, x 1 ( t ) = [ 1 3 − 4 ] e ( − 18 ) t , x 2 ( t ) = [ 3 2 − 5 ] e ( − 11 ) t and x 3 ( t ) = [ 10 3 20 ] e ( 0 ) 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

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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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 35P

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Program Description:Purpose of problem is to find 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 34P

Chapter 5.2, Problem 36P

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Chapter 5 Solutions

EBK DIFFERENTIAL EQUATIONS

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 find 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 find the amount of salt in three water tanks at time t and draw a graph of x_1(t) and

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Program Description:Purpose of problem is to find 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