Explanation Given: The given non-homogeneous system is, X ′ = ( 1 1 1 0 2 3 0 0 5 ) X + ( 1 − 1 2 ) e 4 t . Calculation: For the given system, the coefficient matrix is A = ( 1 1 1 0 2 3 0 0 5 ) and F ( t ) = ( 1 − 1 2 ) e 4 t . Since, the characteristic equation for the matrix A is | ( A − λ I ) | = 0 where I is the Identity matrix. Substitute the values of matrix A and I in above expression. | 1 − λ 1 1 0 2 − λ 3 0 0 5 − λ | = 0 ( 1 − λ ) ( 2 − λ ) ( 5 − λ ) = 0 Therefore, the Eigen values of the above system are λ 1 = 1 , λ 2 = 2 and λ 3 = 5 Now, to obtain the Eigen vector corresponding to the Eigen value λ 1 = 1 . Solve the equation ( A − λ 1 I ) X = 0 where X = ( x 1 y 1 z 1 ) . Substituting the values of A , λ ` 1 and X in the above equation gives, [ ( 1 1 1 0 2 3 0 0 5 ) − ( 1 0 0 0 1 0 0 0 1 ) ] ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) ( 0 1 1 0 1 3 0 0 4 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) Applying the row transformation R 2 → R 2 − R 1 in above system, ( 0 1 1 0 0 2 0 0 4 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) Applying the row transformation R 3 → R 3 − 2 R 2 in above system, ( 0 1 1 0 0 2 0 0 0 ) ⋅ ( x 1 y 1 z 1 ) = ( 0 0 0 ) ( y 1 + z 1 2 z 1 0 ) = ( 0 0 0 ) Solving above equations gives, z 1 = 0 and y 1 = 0 . Assume that x 1 = m for some scalar m . Therefore, the value of ( x 1 y 1 z 1 ) is, ( x 1 y 1 z 1 ) = ( m 0 0 ) = m ( 1 0 0 ) Therefore, the Eigenvector K 1 corresponding to the Eigen value λ 1 = 1 is ( 1 0 0 ) . Now, to obtain the Eigen vector K 2 corresponding to the Eigen value λ 2 = 2 . Solve the equation ( A − λ 2 I ) K 2 = 0 where K 2 = ( x 2 y 2 z 2 ) . Substituting the values of A , λ ` 2 and K 2 in the above equation gives, [ ( 1 1 1 0 2 3 0 0 5 ) − ( 2 0 0 0 2 0 0 0 2 ) ] ⋅ ( x 2 y 2 z 2 ) = ( 0 0 0 ) ( − 1 1 1 0 0 3 0 0 3 ) ⋅ ( x 2 y 2 z 2 ) = ( 0 0 0 ) Applying the row transformation R 3 → R 3 − R 2 in above system, ( − 1 1 1 0 0 3 0 0 0 ) ⋅ ( x 2 y 2 z 2 ) = ( 0 0 0 ) ( − x 2 + y 2 + z 2 3 z 2 0 ) = ( 0 0 0 ) Solving above equations gives, z 2 = 0 and y 2 = x 2 . Assume that x 2 = n for some scalar n which implies y 2 = n . Therefore, the value of ( x 2 y 2 z 2 ) is, ( x 2 y 2 z 2 ) = ( n n 0 ) = n ( 1 1 0 ) Therefore the eigenvector K 2 corresponding to the Eigen value λ 2 = 2 is ( 1 1 0 ) . Now, to obtain the Eigen vector K 3 corresponding to the Eigen value λ 3 = 5 . Solve the equation ( A − λ 3 I ) K 3 = 0 where K 3 = ( x 3 y 3 z 3 )

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

9th Edition

ISBN: 9780357539545

Author: ZILL

Publisher: CENGAGE L

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Chapter 8.3, Problem 7E

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To solve: The given non-homogeneous system using the method of undetermined coefficient.

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Chapter 8.3, Problem 6E

Chapter 8.3, Problem 8E

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Chapter 8 Solutions

WEB ASSIGN FOR ZILL'S DIFFERENTIAL EQUAT

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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