Explanation Given: The given non-homogeneous system is, X ′ = ( 1 3 3 1 ) X + ( − 2 t 2 t + 5 ) . Calculation: For the given system, the coefficient matrix is A = ( 1 3 3 1 ) and F ( t ) = ( − 2 t 2 t + 5 ) . 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 3 3 1 − λ | = 0 ( 1 − λ ) ( 1 − λ ) − 9 = 0 λ 2 − 2 λ − 8 = 0 ( λ − 4 ) ( λ + 2 ) = 0 Therefore, the Eigen values of the above system is λ 1 = 4 and λ 2 = − 2 . Now, to obtain the Eigen vector K 1 corresponding to the Eigen value λ 1 = 4 solve the equation ( A − λ 1 I ) K 1 = 0 where, K 1 = ( x 1 y 1 ) . Substituting the values of A , λ ` 1 and K 1 in the above equation gives, [ ( 1 3 3 1 ) − ( 4 0 0 4 ) ] ⋅ ( x 1 y 1 ) = ( 0 0 ) ( − 3 3 3 − 3 ) ⋅ ( x 1 y 1 ) = ( 0 0 ) Applying the row transformation R 2 → R 2 + R 1 in above system, ( − 3 3 0 0 ) ⋅ ( x 1 y 1 ) = ( 0 0 ) ( − 3 x 1 + 3 y 1 0 ) = ( 0 0 ) Solving above equations gives, y 1 = x 1 Assume that x 1 = n for some scalar n which implies y 1 = n . Therefore, the value of ( x 1 y 1 ) is, ( x 1 y 1 ) = ( n n ) = n ( 1 1 ) Therefore the eigenvector K 1 corresponding to the Eigen value λ 1 = 4 is ( 1 1 ) . 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 ) . Substituting the values of A , λ ` 2 and K 2 in the above equation gives, [ ( 1 3 3 1 ) − ( − 2 0 0 − 2 ) ] ⋅ ( x 2 y 2 ) = ( 0 0 ) ( 3 3 3 3 ) ⋅ ( x 2 y 2 ) = ( 0 0 ) , Applying the row transformation R 2 → R 2 − R 1 in above system, ( 3 3 0 0 ) ⋅ ( x 2 y 2 ) = ( 0 0 ) ( 3 x 2 + 3 y 2 0 ) = ( 0 0 ) Solving above equations gives, y 2 = − x 2 . Assume that x 2 = m for some scalar m which implies y 2 = − m . Therefore, the value of ( x 2 y 2 ) is, ( x 2 y 2 ) = ( m − m ) = m ( 1 − 1 ) Therefore the eigenvector K 2 corresponding to the Eigen value λ 2 = − 2 is ( 1 − 1 ) . The complementary function for the given non homogeneous system is, X c = c 1 K 1 e λ 1 t + c 2 K 2 e λ 2 t . Substitute the value of λ 1 , λ 2 , K 1 and K 2 in above equation, X c = c 1 ( 1 1 ) e 4 t + c 2 ( 1 − 1 ) e − 2 t Since the function F ( t ) is given as F ( t ) = ( − 2 t 2 t + 5 ) F ( t ) = ( − 2 0 ) t 2 + ( 0 1 ) t + ( 0 5 ) Therefore, to obtain the particular solution X p for the given system, assume that X p is given of the form, X p = ( a 3 b 3 ) t 2 + ( a 2 b 2 ) t + ( a 1 b 1 ) , where a 1 , a 2 , a 3 , b 1 , b 2 , b 3 are constants

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 8.3, Problem 3E

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

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Let the universal set be whole numbers 1 through 20 inclusive. That is, U = {1, 2, 3, 4, . . ., 19, 20}. Let A, B, and C be subsets of U. Let A be the set of all prime numbers: A = {2, 3, 5, 7, 11, 13, 17, 19} Let B be the set of all odd numbers: B = {1,3,5,7, . . ., 17, 19} Let C be the set of all square numbers: C = {1,4,9,16}

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Differential Equations with Boundary-Value Problems

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 given non-homogeneous system using the method of undetermined coefficient.

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Chapter 8 Solutions