Explanation Given: The system of differential equations is { d x d t = 2 x + y + 2 z d y d t = 3 x + 6 z d z d t = − 4 x − 3 z . Calculation: The given system of differential equation in matrix notation is as follows: X ′ = ( 2 1 2 3 0 6 − 4 0 − 3 ) X Here, the matrix are defined as A = ( 2 1 2 3 0 6 − 4 0 − 3 ) , X = ( x y z ) and X ′ = ( d x d t d y d t d z d t ) . The characteristic equation for the above matrix A is det ( A − λ I ) = 0 . det ( A − λ I ) = 0 | 2 − λ 1 2 3 0 − λ 6 − 4 0 − 3 − λ | = 0 − 4 ( 6 + 2 λ ) + ( − 3 − λ ) ( ( − λ ) ( 2 − λ ) − 3 ) = 0 ( Expand along row 3 ) − 4 ( 6 + 2 λ ) − ( 3 + λ ) ( − 2 λ + λ 2 − 3 ) = 0 Further, simplify the above equation. − 4 ( 2 ) ( 3 + λ ) − ( 3 + λ ) ( − 2 λ + λ 2 − 3 ) = 0 ( 3 + λ ) { − 8 + 2 λ − λ 2 + 3 } = 0 − ( 3 + λ ) { 8 − 2 λ + λ 2 − 3 } = 0 ( 3 + λ ) ( λ 2 − 2 λ + 5 ) = 0 Solve the above characteristic equation as follows: 3 + λ = 0 or λ 2 − 2 λ + 5 = 0 Linear equation gives the value of λ = − 3 . The quadratic equation λ 2 − 2 λ + 5 = 0 is solved as follows: λ = − ( − 2 ) ± ( − 2 ) 2 − 4 ( 1 ) ( 5 ) 2 ( 1 ) = 2 ± 4 − 20 2 = 2 ± 4 i 2 = 1 ± 2 i Therefore, the Eigen values of the coefficient matrix are λ 1 = − 3 , λ 2 = 1 + 2 i and λ 3 = 1 − 2 i . Now, to obtain the Eigenvector corresponding to the Eigen value λ 1 = − 3 , substitute the value of Eigen value in the equation ( A − λ I ) K = 0 . ( A − ( − 3 ) I ) K = 0 ( 5 1 2 3 3 6 − 4 0 0 ) ( x y z ) = ( 0 0 0 ) Here, the matrix equation is equivalent to linear equations as shown below. 5 x + y + 2 z = 0 ...... ( 1 ) 3 x + 3 y + 6 z = 0 ...... ( 2 ) − 4 x = 0 ...... ( 3 ) From equation (3), the value for x is 0 . Substitute 0 for x in equation (1) and (2). y + 2 z = 0 ...... ( 4 ) 3 y + 6 z = 0 ...... ( 5 ) It is observed that both the above equations are equivalent. Therefore, take z = 1 in the equation (4) and obtain the value of y = − 2 . So, the first Eigenvector corresponds to the Eigen value λ 1 = − 3 is as follows: K 1 = ( 0 − 2 1 ) Therefore, the first corresponding solution of the differential equation is as follows: X 1 = K 1 e λ 1 t X 1 = ( 0 − 2 1 ) e − 3 t Similarly, to obtain the Eigenvector corresponding to the Eigen value λ 2 = 1 + 2 i , substitute the value of Eigen value in the equation ( A − λ I ) K = 0 . ( A − ( 1 + 2 i ) I ) K = 0 ( 2 − ( 1 + 2 i ) 1 2 3 − ( 1 + 2 i ) 6 − 4 0 − 3 − ( 1 + 2 i ) ) ( x y z ) = ( 0 0 0 ) ( 1 − 2 i 1 2 3 − ( 1 + 2 i ) 6 − 4 0 − 4 − 2 i ) ( x y z ) = ( 0 0 0 ) Here, the matrix equation is equivalent to linear equations as shown below. ( 1 − 2 i ) x + y + 2 z = 0 ...... ( 6 ) 3 x − ( 1 + 2 i ) y + 6 z = 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8.2, Problem 42E

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The general solution of the system of differential equations {dxdt=2x+y+2zdydt=3x+6zdzdt=−4x−3z.

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Chapter 8.2, Problem 41E

Chapter 8.2, Problem 43E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - In Problems 710 write the given linear system...Ch. 8.1 - Prob. 8E Ch. 8.1 - In Problems 16 write the given linear system in...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - In Problems 1116 verify that the vector X is a...Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - In Problems 1720 the given vectors are solutions...Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - In Problems 112 find the general solution of the...Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Distinct Real Eigenvalues In Problems 112 find the...Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - In Problem 27 of Exercises 4.9 you were asked to...Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Prob. 44E Ch. 8.2 - In Problems 3546 find the general solution of the...Ch. 8.2 - Prob. 46E Ch. 8.2 - In Problems 47 and 48 solve the given...Ch. 8.2 - Prob. 48E Ch. 8.2 - The system of mixing tanks shown in Figure 8.2.7...Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 53E Ch. 8.2 - Show that the 5 5 matrix...Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - In Problems 18 use the method of undetermined...Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Consider the large mixing tanks shown in Figure...Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - The system of differential equations for the...Ch. 8.3 - Prob. 36E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - In problem 58 use (1) use to find the general...Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - In problem 1518 use the method of Example 2 to...Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8 - fill in the blanks. 1. The vector X=k(45) is a...Ch. 8 - fill in the blanks. The vector...Ch. 8 - Consider the linear system X=(466132143)X. Without...Ch. 8 - Consider the linear system X = AX of two...Ch. 8 - In Problems 514 solve the given linear system. 5....Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE

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The general solution of the system of differential equations { d x d t = 2 x + y + 2 z d y d t = 3 x + 6 z d z d t = − 4 x − 3 z .

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The general solution of the system of differential equations {dxdt=2x+y+2zdydt=3x+6zdzdt=−4x−3z.

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Chapter 8 Solutions