(a) Explanation Given: The matrix system of differential equation is as follows: d d t ( i 2 i 3 ) = ( − R 1 L 1 − R 1 L 1 − R 1 L 2 − ( R 1 + R 2 ) L 2 ) ( i 2 i 3 ) + ( E L 1 E L 2 ) The value of R 1 , R 2 , L 1 and L 2 are as follows: R 1 = 2 Ω R 2 = 3 Ω L 1 = 1 h L 2 = 1 h The initial value problem is i 2 ( 0 ) = 0 and i 3 ( 0 ) = 0 . Calculation: The matrix system of differential equation is as follows: d d t ( i 2 i 3 ) = ( − R 1 L 1 − R 1 L 1 − R 1 L 2 − ( R 1 + R 2 ) L 2 ) ( i 2 i 3 ) + ( E L 1 E L 2 ) … … ( 1 ) It is given that the value of R 1 , R 2 , L 1 , E and L 2 are R 1 = 2 Ω , R 2 = 3 Ω , L 1 = 1 h , L 2 = 1 h and E = 60 V . Substitute the value of R 1 , R 2 , L 1 , E and L 2 in the equation ( 1 ) . d d t ( i 2 i 3 ) = ( − 2 − 2 − 2 − 5 ) ( i 2 i 3 ) + ( 60 60 ) The above equation is in the form as X ′ = A X + F ( i ) , where A = ( − 2 − 2 − 2 − 5 ) and F ( i ) = ( 60 60 ) . The determinant of A − λ I is calculated as follows: | A − λ I | = 0 | − 2 − λ − 2 − 2 − 5 − λ | = 0 ( − 2 − λ ) ( − 5 − λ ) − 4 = 0 λ 2 + 7 λ + 6 = 0 ( λ + 1 ) ( λ + 6 ) = 0 Further solve the above equation. The first value of λ is calculated as follows: λ + 1 = 0 λ = − 1 ( Subtract 1 ) The second value of λ is calculated as follows: λ + 6 = 0 λ = − 6 ( Subtract 6 ) The Eigen value for the matrix A is λ 1 = − 1 and λ 2 = − 6 . The Eigen vector for λ 1 = − 1 is calculated as follows: ( − 2 − λ − 2 − 2 − 5 − λ ) ( x 1 x 2 ) = ( 0 0 ) Substitute − 1 for λ in the above equation. ( − 1 − 2 − 2 − 4 ) ( x 1 x 2 ) = ( 0 0 ) The equations formed are as follows: − x 1 − 2 x 2 = 0 − 2 x 1 − 4 x 2 = 0 From the above two it is concluded that x 1 = − 2 x 2 . Consider the value of x 2 as 1 then the value of x 1 is − 2 . Therefore, the Eigen vector for the Eigen value λ 1 = − 1 is as follows: X 1 = ( − 2 1 ) The Eigen vector for λ 1 = − 6 is calculated as follows: ( − 2 − λ − 2 − 2 − 5 − λ ) ( x 1 x 2 ) = ( 0 0 ) Substitute − 6 for λ in the above equation. ( 4 − 2 − 2 1 ) ( x 1 x 2 ) = ( 0 0 ) The equations formed are as follows: 4 x 1 − 2 x 2 = 0 − 2 x 1 − x 2 = 0 From the above two it is concluded that x 2 = 2 x 1 . Consider the value of x 1 as 1 then the value of x 2 is 2 . Therefore, the Eigen vector for the Eigen value λ = − 6 is as follows: X 1 = ( 1 2 ) The solution of the homogenous part of the given equation is as follows: X c = c 1 e − t ( − 2 1 ) + c 2 e − 6 t ( 1 2 ) The given system is in the form as X ′ = A X + F ( i ) , where A = ( − 2 − 2 − 2 − 5 ) and F ( i ) = ( 60 60 ) (b) To determine The value of the current i 1 ( t ) in the given diagram of the circuit.

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 8.3, Problem 12E

(a)

To determine

The solution of the system of differential equation ddt(i2i3)=(−R1L1−R1L1−R1L2−(R1+R2)L2) (i2i3)+(EL1EL2) by the method of undetermined coefficients.

(b)

To determine

The value of the current i1(t) in the given diagram of the circuit.

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Chapter 8.3, Problem 11E

Chapter 8.3, Problem 13E

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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 solution of the system of differential equation d d t ( i 2 i 3 ) = ( − R 1 L 1 − R 1 L 1 − R 1 L 2 − ( R 1 + R 2 ) L 2 ) ( i 2 i 3 ) + ( E L 1 E L 2 ) by the method of undetermined coefficients.

Solution Summary: The author explains the solution of the system of differential equation dt(li_2) by the method of undetermined coefficients.