Explanation Given: The given non-homogeneous system is, X ′ = ( 3 − 1 − 1 1 1 − 1 1 − 1 1 ) X + ( 0 t 2 e t ) Calculation: The system is given in the form X ′ = A X + F ( t ) (1). Consider, the characteristic equation of the coefficient matrix A as | A − λ I | = 0 Now, the characteristic equation of the coefficient matrix A . is | ( A − λ I ) | = 0 | 3 − λ − 1 − 1 1 1 − λ − 1 1 − 1 1 − λ | = 0 ( 3 − λ ) [ ( 1 − λ ) 2 − 1 ] + 1 [ 1 − λ + 1 ] − 1 [ − 1 − ( 1 − λ ) ] = 0 ( 3 − λ ) [ λ 2 − 2 λ + 1 − 1 ] + [ 2 − λ ] − ( λ − 2 ) = 0 Simplify the above obtained expression, ( 3 − λ ) [ λ 2 − 2 λ ] + 2 ( 2 − λ ) = 0 ( 2 − λ ) [ ( 3 − λ ) λ + 2 ] = 0 ( 2 − λ ) [ λ 2 − 3 λ + 2 ] = 0 ( 2 − λ ) ( λ − 1 ) ( λ − 2 ) = 0 Simplify the above obtained expression, λ = 1 , 2 , 2 So, Eigen values are λ 1 = 2 , λ 2 = 2 , λ 3 = 1 . Now evaluating the Eigen vector for λ 1 = 2 , ( A − 2 I 0 ) = ( 1 − 1 − 1 0 1 − 1 − 1 0 1 − 1 − 1 0 ) On solving, it gives K 1 − K 2 − K 3 = 0 Which on further evaluation becomes K 1 = K 3 + K 2 . There are three unknown variables in one equation, so first write the general Eigen vector and then break it into two vectors which gives two Eigen vectors for repeated Eigen values. So, general Eigen vector is K = [ K 3 + K 2 K 2 K 3 ] . Now, separate the Eigen vectors as, [ K 3 + K 2 K 2 K 3 ] = K 2 [ 1 1 0 ] + K 3 [ 1 0 1 ] So the corresponding Eigen vectors for repeated Eigen values are K 1 = ( 1 − 1 0 ) , K 2 = ( 1 0 1 ) . Thus, the corresponding solution vector is X 1 = ( 1 − 1 0 ) e 2 t and X 2 = ( 1 0 1 ) e 2 t . Now evaluating the Eigen vector for λ 3 = 1 , ( A − I 0 ) = ( 2 − 1 − 1 0 1 0 − 1 0 1 − 1 0 0 ) On solving, it gives 2 K 1 − K 2 − K 3 = 0 K 1 − K 3 = 0 K 1 − K 2 = 0 Which on further evaluation becomes, K 1 − K 3 = 0 K 1 = K 2 K 1 = K 3 . So the corresponding Eigen vector is K 3 = ( 1 1 1 ) . Thus the corresponding solution vector is X 3 = ( 1 1 1 ) e t . Consider the coefficient matrix A having real Eigen values of homogeneous system and K be an Eigen vector and X 1 , X 2 and X 3 be the solution vectors for the corresponding Eigen value, then the value of complementary function is X c = C 1 X 1 + C 2 X 2 + C 3 X 3 Substituting the value of X 1 , X 2 and X 3 in above expression gives, X c = C 1 ( 1 1 0 ) e t + C 2 ( 1 1 0 ) e 2 t + C 3 ( 1 0 1 ) e 2 t Now, the entries in X 1 form the first column of solution matrix ϕ ( t ) , the entries in X 2 form the second column, the entries in X 3 form the third column of ϕ ( t ) It means, the value of ϕ ( t ) is ϕ ( t ) = ( e t e 2 t e 2 t e t e 2 t 0 e t 0 e 2 t ) Now, evaluating determinant of the above matrix ϕ ( t ) gives det [ ϕ ( t ) ] = | e t e 2 t e 2 t e t e 2 t 0 e t 0 e 2 t | = e t ( 0 − e 4 t ) + e 2 t ( e 3 t − e 3 t ) = e 5 t Now, evaluating co-factor matrix of the above matrix ϕ ( t ) gives C = ( e 4 t − e 3 t − e 3 t − e 4 t 0 e 3 t e 4 t e 3 t 0 ) Now, the transpose of the co-factor matrix gives Adj ϕ ( t )

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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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 by using variation of parameters.

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