Explanation Given: The linear system is X ′ = ( 1 2 − 1 2 1 ) X + ( 0 e t tan t ) . Calculation: The system is in the form X ′ = A X + F ( t ) . Write the characteristic equation of the coefficient matrix A . ( A − λ I ) = ( 1 − λ 2 − 1 2 1 − λ ) For the Eigen values, determinant of the coefficient matrix equate to zero. Det ( A − λ I ) = | 1 − λ 2 − 1 2 1 − λ | ( 2 − λ ) 2 + 1 = 0 λ 2 − 2 λ + 2 = 0 The values of λ is calculated as follows: λ = 2 ± 4 − 4 ⋅ 2 2 λ = 1 ± i λ 1 = 1 + i or λ 2 = 1 − i where λ 1 , λ 2 are the two Eigen values. For λ 1 = 1 + i , the Eigen vector is given below. ( A − ( 1 + i ) I | 0 ) = ( − i 2 0 − 1 2 − i 0 ) Therefore, the value is − i K 1 + 2 K 2 = 0 From the above expression K 2 = i 2 K 1 . The Corresponding Eigen vector is K 1 = ( 2 i ) . Consider A is the coefficient matrix having real values of homogeneous system and K is an Eigen vector corresponding the complex Eigen value λ 1 = 1 + i then the solution is X = K e λ 1 t and this solution will come in the form of X = X 1 + X 2 . X = ( 2 i ) e ( 1 + i ) t = ( 2 i ) e t ( cos t + i sin t ) = ( 2 cos t − sin t ) e t + i ( 2 sin t cos t ) e t The solution vector of the system is given below. X 1 = ( 2 cos t − sin t ) e t , X 2 = ( 2 sin t cos t ) e t The solution vectors can be written as follows: ϕ ( t ) = ( 2 cos t ⋅ e t 2 sin t ⋅ e t − sin t ⋅ e t cos t ⋅ e t ) Det ϕ ( t ) = ( 2 cos t ⋅ e t ) ⋅ ( cos t ⋅ e t ) − ( − sin t ⋅ e t ) ⋅ ( cos t ⋅ e t ) = 2 e t cos 2 t + 2 e 2 t sin 2 t = 2 e 2 t Determination of ϕ − 1 ( t ) is calculated as follows: ϕ − 1 ( t ) = 1 ϕ − 1 ( t ) ( cos t ⋅ e t − 2 sin t ⋅ e t sin t ⋅ e t 2 cos t ⋅ e t ) ϕ − 1 ( t ) = ( 1 2 e − t cos t − e − t sin t 1 2 e − t sin t ⋅ e − t cos t ) Express the particular solution X p as given below. X p = ϕ ( t ) ∫ ϕ − 1 ( t ) F ( t ) d t

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, Problem 12RE

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The solution of the linear system X′=(12−121)X+(0ettant).

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Chapter 8, Problem 11RE

Chapter 8, Problem 13RE

Chapter 8 Solutions

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 solution of the linear system X ′ = ( 1 2 − 1 2 1 ) X + ( 0 e t tan t ) .

Solution Summary: The author explains the linear system Xprime, which is underset_x=1left.

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The solution of the linear system X′=(12−121)X+(0ettant).

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Chapter 8 Solutions