Explanation Given: The differential system in matrix notation is X ′ = ( 4 0 1 0 6 0 − 4 0 4 ) X . Calculation: The differential system X ′ = ( 4 0 1 0 6 0 − 4 0 4 ) X is equivalent to X ′ = A X . Here, the matrix are defined as A = ( 4 0 1 0 6 0 − 4 0 4 ) , X = ( x y z ) and X ′ = ( d x d t d y d t d z d t ) . The characteristic equation for the above matrix A is det ( A − λ I ) = 0 . det ( A − λ I ) = 0 | 4 − λ 0 1 0 6 − λ 0 − 4 0 4 − λ | = 0 ( 4 − λ ) [ ( 6 − λ ) ( 4 − λ ) ] + 4 ( 6 − λ ) = 0 ( Expand along row 1 ) ( 6 − λ ) ( λ 2 − 8 λ + 20 ) = 0 Solve the above characteristic equation as follows: 6 − λ = 0 or λ 2 − 8 λ + 20 = 0 Linear equation gives the value of λ = 6 . The quadratic equation λ 2 − 8 λ + 20 = 0 is solved as follows: λ = − ( − 8 ) ± ( − 8 ) 2 − 4 ( 1 ) ( 20 ) 2 ( 1 ) = 8 ± 64 − 80 2 = 8 ± 4 i 2 = 4 ± 2 i Therefore, the Eigen values of the coefficient matrix are λ 1 = 6 , λ 2 = 4 + 2 i and λ 3 = 4 − 2 i . Now, to obtain the Eigenvector corresponding to the Eigen value λ 1 = 6 , substitute the value of Eigen value in the equation ( A − λ I ) K = 0 . ( A − ( 6 ) I ) K = 0 ( 4 − 6 0 1 0 6 − 6 0 − 4 0 4 − 6 ) ( x y z ) = ( 0 0 0 ) ( − 2 0 1 0 0 0 − 4 0 − 2 ) ( x y z ) = ( 0 0 0 ) Simplify the above matrix equation by the use of Gauss Jordan elimination method. ( − 4 0 − 2 0 0 0 − 2 0 1 ) ( x y z ) = ( 0 0 0 ) [ R 1 ↔ R 3 ] ( − 4 0 − 2 0 0 0 0 0 2 ) ( x y z ) = ( 0 0 0 ) [ R 3 → R 3 − 1 2 R 1 ] ( − 4 0 0 0 0 0 0 0 2 ) ( x y z ) = ( 0 0 0 ) [ R 1 → R 1 + R 3 ] ( − 4 0 0 0 0 0 0 0 2 ) ( x y z ) = ( 0 0 0 ) [ R 1 → R 1 + R 3 ] Further, simplify the above matrix equation. ( 1 0 0 0 0 0 0 0 1 ) ( x y z ) = ( 0 0 0 ) [ R 1 → 1 − 4 R 1 R 3 → 1 2 R 3 ] Solve the above matrix equation and it gives x = 0 and z = 0 and y is independent. Therefore, take y = 1 and the first Eigenvector corresponds to the Eigen value λ 1 = 6 is as follows: K 1 = ( 0 1 0 ) Therefore, the first corresponding solution of the differential equation is as follows: X 1 = K 1 e λ 1 t X 1 = ( 0 1 0 ) e 6 t Similarly, to obtain the Eigenvector corresponding to the Eigen value λ 2 = 4 + 2 i , substitute the value of Eigen value in the equation ( A − λ I ) K = 0 . ( A − ( 4 + 2 i ) I ) K = 0 ( 4 − ( 4 + 2 i ) 0 1 0 6 − ( 4 + 2 i ) 0 − 4 0 4 − ( 4 + 2 i ) ) ( x y z ) = ( 0 0 0 ) ( − 2 i 0 1 0 2 − 2 i 0 − 4 0 − 2 i ) ( x y z ) = ( 0 0 0 ) Simplify the above matrix equation by the use of Gauss Jordan elimination method

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

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 8.2, Problem 44E

To determine

The general solution of the differential system X′=(401060−404)X.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.2, Problem 43E

Chapter 8.2, Problem 45E

Students have asked these similar questions

Analyze the residuals of a linear regression model and select the best response.Criteria is simple evaluation of possible indications of an exponential model vs. linear model) no, the residual plot does not show a curve yes, the residual plot does not show a curve yes, the residual plot shows a curve no, the residual plot shows a curve I selected: yes, the residual plot shows a curve and it is INCORRECT. Can u help me understand why?

You have been hired as an intern to run analyses on the data and report the results back to Sarah; the five questions that Sarah needs you to address are given below. please do it step by step on excel Does there appear to be a positive or negative relationship between price and screen size? Use a scatter plot to examine the relationship. Determine and interpret the correlation coefficient between the two variables. In your interpretation, discuss the direction of the relationship (positive, negative, or zero relationship). Also discuss the strength of the relationship. Estimate the relationship between screen size and price using a simple linear regression model and interpret the estimated coefficients. (In your interpretation, tell the dollar amount by which price will change for each unit of increase in screen size). Include the manufacturer dummy variable (Samsung=1, 0 otherwise) and estimate the relationship between screen size, price and manufacturer dummy as a multiple…

(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).

Chapter 8 Solutions

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The general solution of the differential system X ′ = ( 4 0 1 0 6 0 − 4 0 4 ) X .

Solution Summary: The author explains how the differential system Xprime is equivalent to the matrix A=(cc

Videos

The general solution of the differential system X′=(401060−404)X.

Want to see the full answer?

Chapter 8 Solutions