Explanation Given: Tank A and tank B holds 50 L of liquid, are interconnected by pipes. The liquid flows from tank A into tank B at a rate of 4 L/min and from B into A at a rate of 1 L/min. Calculation: The rate of change of salt in tank A is, d x 1 ( t ) d t = 0.8 + x 2 ( t ) 50 − x 1 ( t ) 50 − 4 x 1 ( t ) 50 = − x 1 ( t ) 10 + x 2 ( t ) 50 + 0.8 … ( 1 ) The rate of change of salt in tank B is, d x 2 ( t ) d t = 0.1 + 4 x 1 ( t ) 50 − x 2 ( t ) 50 − 4 x 2 ( t ) 50 = 4 x 1 ( t ) 50 − x 2 ( t ) 10 + 0.1 … ( 2 ) The matrix form of Equation (1) and Equation (2) is, [ x ′ 1 x ′ 2 ] = [ − 1 10 1 50 4 50 − 1 10 ] [ x 1 x 2 ] + [ 0.8 0.1 ] Consider the matrix, A = [ − 1 10 1 50 4 50 − 1 10 ] The characteristic Equation of matrix A is. | − 1 10 − r 1 50 4 50 − 1 10 − r | = 0 ( − 1 10 − r ) ( − 1 10 − r ) − ( 4 50 ) ( 1 50 ) = 0 r = − 7 50 , − 3 50 The Eigen vector corresponding to r = − 7 50 are, u 1 = [ − 1 2 1 ] The Eigen vector corresponding to r = − 3 50 are, u 2 = [ 1 2 1 ] The solution of the system is, x h = c 1 [ − 1 2 e − 7 50 t e − 7 50 t ] + c 2 [ 1 2 e − 3 50 t e − 3 50 t ] = [ − 1 2 e − 7 50 t 1 2 e − 3 50 t e − 7 50 t e − 3 50 t ] [ c 1 c 2 ] … ( 3 ) The fundamental matrix is, Ω = [ − 1 2 e − 7 50 t 1 2 e − 3 50 t e − 7 50 t e − 3 50 t ] The inverse of matrix is, Ω − 1 = 1 7 [ e − 5 t − 4 e 2 t e − 5 t 3 e 2 t ] = 1 2 e 3 t 50 [ − 2 e 2 t 25 e 2 t 25 2 1 ] The particular solution of system is, x p = Ω ∫ 0 t Ω − 1 f ( s ) d s … ( 4 ) Substitute 1 2 e 3 s 50 [ − 2 e 2 s 25 e 2 s 25 2 1 ] for Ω − 1 , [ − 1 2 e − 7 50 t 1 2 e − 3 50 t e − 7 50 t e − 3 50 t ] for Ω and [ 0.8 0.1 ] for in f ( s ) Equation (4). x p = [ − 1 2 e − 7 50 t 1 2 e − 3 50 t e − 7 50 t e − 3 50 t ] ∫ 0 t 1 2 e 3 s 50 [ − 2 e 2 s 25 e 2 s 25 2 1 ] [ 0.8 0.1 ] d s = [ − 1 2 e − 7 50 t 1 2 e − 3 50 t e − 7 50 t e − 3 50 t ] ∫ 0 t [ − 0

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 9.7, Problem 34E

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The tank having greater concentration and the limiting concentration in tank A and B.

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The tank having greater concentration and the limiting concentration in tank A and B.

Solution Summary: The author calculates the rate of change of salt in tank A and tank B.

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The tank having greater concentration and the limiting concentration in tank A and B.

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