Program Explanation Given information: The volume of tank 1 is 50 gal and volume of tank 2 is 25 gal . The value of r is 10 gal / min , x 1 ( 0 ) = 15 ( l b ) and x 2 ( 0 ) = 0 . Explanation: Consider the rate equation for the tank 1 as follows. d x 1 d t = − r V 1 x 1 …… (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) Substitute 10 for r and 50 gal for V 1 in equation (1). d x 1 d t = − 10 50 x 1 = − 0.2 x 1 + 0 x 2 Substitute 10 for r , 25 gal for V 2 and 50 gal for V 1 in equation (2). d x 1 d t = 10 50 x 1 − 10 25 x 2 = 0.2 x 1 − 0.4 x 2 Consider matrix form of the system of equation as A. A = [ − 0.2 0 0.2 − 0.4 ] The characteristics equation of the coefficient matrix is, [ − 0.2 − λ 0 0.2 − 0.4 − λ ] = 0 ( − 0.2 − λ ) ( − 0.4 − λ ) = 0 λ = − 0.2 or λ = − 0.4 The Eigen vector of the coefficient matrix A is in the form as, [ − 0.2 − λ 0 0.2 − 0.4 − λ ] [ a b ] = [ 0 0 ] The associated Eigen vector with the matrix Ais v = [ a b ] T . Substitute − 0.2 for λ in coefficient matrix, [ − 0.2 + 0.2 0 0.2 − 0.4 + 0.2 ] [ a 1 b 1 ] = [ 0 0 ] [ 0 0 0.2 − 0.2 ] [ a 1 b 1 ] = [ 0 0 ] After solving the equation the value of a 1 and b 1 can be obtained as, a 1 = 1 b 1 = 1 Therefore, the associated Eigen vector is shown below. v 1 = [ 1 1 ] Substitute − 0.4 for λ in coefficient matrix, [ − 0.2 + 0.4 0 0.2 − 0.4 + 0.4 ] [ a 2 b 2 ] = [ 0 0 ] [ 0.2 0 0.2 0 ] [ a 2 b 2 ] = [ 0 0 ] After solving the equation the value of a 1 and b 1 can be obtained as, a 2 = 0 b 2 = 1 Therefore, the associated Eigen vector is shown below

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Linear Programming Concepts

Linear programming is a famous mathematical modeling tool for determining the best distribution of scarce resources among competing demands. Machinery, time, personnel, raw materials, money, and space are all resources. It is used to find the most optima…

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Chapter 5.2, Problem 27P

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Program Description:Purpose of problem is to obtain the maximum amount of salt ever in tank 2 and also construct a graph of x1(t) and y1(t) .

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Chapter 5.2, Problem 26P

Chapter 5.2, Problem 28P

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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

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Program Description: Purpose of problem is to obtain the maximum amount of salt ever in tank 2 and also construct a graph of x 1 ( t ) and y 1 ( t ) .

Solution Summary: The author explains that the problem is to obtain the maximum amount of salt ever in tank 2 and construct a graph of x_1(t)andy

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Program Description:Purpose of problem is to obtain the maximum amount of salt ever in tank 2 and also construct a graph of x1(t) and y1(t) .

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Chapter 5 Solutions