Explanation Given: The system is d y d t = 2 x − y d y d t = 3 x − 2 y + 4 t . Calculation: Express the linear system d y d t = 2 x − y and d y d t = 3 x − 2 y + 4 t in the matrix form. X ′ = ( 2 − 1 3 − 2 ) X + ( 0 4 ) t ...... ( 1 ) The homogeneous system from the matrix form is X ′ = ( 2 − 1 3 − 2 ) X . The characteristic equation of the coefficient matrix can be expressed as follows: ( A − λ I ) = ( 2 − 1 3 − 2 ) − λ ( 1 0 0 1 ) = ( 2 − 1 3 − 2 ) − ( λ 0 0 λ ) = ( 2 − λ − 1 3 − 2 − λ ) Express the determinant of the coefficient matrix and equate to zero to obtain the Eigen values. | ( A − λ I ) | = | 2 − λ − 1 3 − 2 − λ | ( 2 − λ ) ( − 2 − λ ) + 3 = 0 λ 2 − 1 = 0 λ = ± 1 Hence, from the above results, the two Eigen values are λ 1 = + 1 and λ 2 = − 1 . For λ 1 = 1 , the components of Eigen vector are calculated as follows: | 2 − λ − 1 3 − 2 − λ | = | 2 − 1 − 1 3 − 2 − 1 | ( λ 1 = 1 ) = | 1 − 1 3 − 3 | K 1 − K 2 = 0 3 K 1 − 3 K 2 = 0 where, K 1 , K 2 are the components of the Eigen vector. From the above expression K 2 = K 1 . Therefore, the corresponding Eigen vector is K 1 = ( 1 1 ) . For λ 2 = − 1 , the components of Eigen vector are calculated as follows: | 2 − λ − 1 3 − 2 − λ | = | 2 + 1 − 1 3 − 2 + 1 | ( λ 1 = − 1 ) = | 3 − 1 3 − 1 | 3 K 1 − K 2 = 0 3 K 1 − K 2 = 0 From the above expression K 2 = 3 K 1 . Therefore, the corresponding Eigen vector is K = ( 1 3 ) . Hence, the solution vectors of the given solution are X 1 = ( 1 1 ) e t , X 2 = ( 1 3 ) e − t . The solution vectors in the form of the matrix ϕ ( t ) can be expressed as follows: ϕ ( t ) = ( e t e − t e t 3 e − t ) Express the determinant of the matrix ϕ ( t )

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8.3, Problem 14E

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The non-homogeneous system dydt=2x−y and dydt=3x−2y+4t using variation of parameter.

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(a) (b) (c) (d) de unique? Answer the following questions related to the linear system x + y + z = 2 x-y+z=0 2x + y 2 3 rewrite the linear system into the matrix-vector form A = 5 Fuse elementary row operation to solve this linear system. Is the solution use elementary row operation to find the inverse of A and then solve the linear system. Verify the solution is the same as (b). give the null space of matrix A and find the dimension of null space. give the column space of matrix A and find the dimension of the column space of A (Hint: use Rank-Nullity Theorem).

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[) Hwk 29 SUBMIT ANSWEK Hwk 30 - (MA 244-03) (SP25) || X - Mind Tap Cengage Learning ☑ MA244-03_Syllabus_Spring, 20 × b Answered: [) 90% Hwk 29 Hwk X Rotation of Axes Example - Elimi X + https://www.webassign.net/web/Student/Assignment-Responses/last?dep=36606609 B שי 90% 2. [-/3 Points] DETAILS MY NOTES LARLINALG8 7.4.003. Use the age transition matrix L and age distribution vector X1 to find the age distribution vectors X2 and x3. 0 34 x2 = X3 = L = ↓ ↑ 1 0 0 x1 = 1 0 0 2 20 20 20 Then find a stable age distribution vector. x = t ↓ 1 Need Help? Read It SUBMIT ANSWER 3. [-/3 Points] DETAILS MY NOTES LARLINALG8 7.4.004. Use the age transition matrix L and age distribution vector X1 to find the age distribution vectors x2 and ×3. ill { ASK YOUR TEACHER PRACTICE ANOTHER ASK YOUR TEACHER PRACTICE ANOTHER

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The non-homogeneous system d y d t = 2 x − y and d y d t = 3 x − 2 y + 4 t using variation of parameter.

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The non-homogeneous system dydt=2x−y and dydt=3x−2y+4t using variation of parameter.

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