Explanation Given: The given system is: X ′ ( t ) = [ 3 − 2 1 2 − 1 1 − 4 4 1 ] X ( t ) Procedure used: Suppose n × n constant matrix A has n linearly independent eigenvectors u 1 , u 2 , ⋯ , u n . The eigenvalues can be finding by using characteristics equation | A − r I | = 0 . Here, r denotes the eigenvalues. The corresponding the vector of the eigenvalues are called eigenvectors. It can be found by substitute the eigenvalues in the formula ( A − r I ) u = 0 . If the eigenvalues are repeated then the solution is: X 1 ( t ) = e r t u 1 , X 2 ( t ) = t e r t u 1 + e r t u 2 and X 3 ( t ) = t 2 2 e r t u 1 + t e r t u 2 + e r t u 3 The general solution is; X ( t ) = X 1 ( t ) + X 2 ( t ) + X 3 ( t ) Calculation: Consider the given matrix: A = [ 3 − 2 1 2 − 1 1 − 4 4 1 ] Substitute the value of matrix A in the characteristic equation and solve for r : | [ 3 − 2 1 2 − 1 1 − 4 4 1 ] − r [ 1 0 0 0 1 0 0 0 1 ] | = 0 | [ 3 − 2 1 2 − 1 1 − 4 4 1 ] − [ r 0 0 0 r 0 0 0 r ] | = 0 | [ 3 − r − 2 1 2 − 1 − r 1 − 4 4 1 − r ] | = 0 ( 3 − r ) ⋅ | − 1 − r 1 4 1 − r | + 2 ⋅ | 2 1 − 4 1 − r | + 1. | 2 − 1 − r − 4 4 | = 0 Further simplify as: ( 3 − r ) ⋅ [ ( − 1 − r ) ( 1 − r ) − 4 ] + 2 ⋅ [ 2 ( 1 − r ) + 4 ] + 1. [ 8 − 4 ( 1 + r ) ] = 0 ( 3 − r ) ⋅ [ − ( 1 − r 2 ) − 4 ] + 2 ⋅ [ 2 − 2 r + 4 ] + [ 8 − 4 − 4 r ] = 0 ( 3 − r ) ⋅ [ r 2 − 5 ] + 12 − 4 r + 4 − 4 r = 0 3 r 2 − 15 − r 3 + 5 r + 16 − 8 r = 0 Further simplify as: − r 3 + 3 r 2 − 3 r + 1 = 0 − ( r − 1 ) 3 = 0 The factor is: ( r − 1 ) 3 = 0 r = 1 The eigenvalues are r 1 , 2 , 3 = 1 . Therefore, the eigenvalues are repeated. Find the eigenvector corresponding to eigenvalue r 1 = 1 by using the formula ( A − r I ) u = 0 :

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.5, Problem 38E

To determine

To find:

The general solution for the given system.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.5, Problem 37E

Chapter 9.5, Problem 39E

Students have asked these similar questions

Please help me with these questions. I am having a hard time understanding what to do. Thank you

3) roadway Calculate the overall length of the conduit run sketched below. 2' Radius 8' 122-62 Sin 30° = 6/H 1309 16.4%. 12' H= 6/s in 30° Year 2 Exercise Book Page 4 10 10 10 fx-300MS S-V.PA Topic 1

© © Q Tue 7 Jan 10:12 pm myopenmath.com/assess2/?cid=253523&aid=17... ookmarks 吕 Student Account... 8 Home | Participant... 001st Meeting with y... E F D c G B H I A J P K L N M Identify the special angles above. Give your answers in degrees. A: 0 B: 30 C: 45 D: 60 E: 90 > १ F: 120 0 G: H: 1: 180 0 J: K: L: 240 0 Next- M: 270 0 0: ZÖÄ N: 300 0 Aa zoom P: Question Help: Message instructor MacBook Air Ο O Σ >> | All Bookmarks

Chapter 9 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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 for the given system.

Solution Summary: The author explains the general solution for the given system. The eigenvalues can be found by using characteristics equation left|A-r

Videos

To find: The general solution for the given system.

Want to see the full answer?

Chapter 9 Solutions

To find:

The general solution for the given system.