Explanation 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 . The general solution is, x ( t ) = c 1 e r 1 t u 1 + c 2 e r 2 t u 2 . Calculation: Consider the given system, t x ′ ( t ) = [ 1 3 − 1 5 ] x ( t ) , t > 0 . Here, A = [ 1 3 − 1 5 ] Let t = e u and t x ′ ( t ) = x ′ ( u ) x ′ ( u ) = [ 1 3 − 1 5 ] x ( u ) , t > 0 Substitute the value of matrix A in the characteristic equation and solve for r : | [ 1 3 − 1 5 ] − r [ 1 0 0 1 ] | = 0 | [ 1 3 − 1 5 ] − [ r 0 0 r ] | = 0 | [ 1 − r 3 − 1 5 − r ] | = 0 ( 1 − r ) ( 5 − r ) + 3 = 0 Further simplify as: 5 − r − 5 r + r 2 + 3 = 0 r 2 − 6 r + 8 = 0 r 2 − 4 r − 2 r + 8 = 0 ( r − 4 ) ( r − 2 ) = 0 Equate first factor to 0. ( r − 4 ) = 0 r = 4 Equate second factor to 0. ( r − 2 ) = 0 r = 2 The eigenvalues are r 1 = 2 , r 2 = 4 . Find the eigenvector corresponding to eigenvalue r 1 = 2 by using the formula ( A − r I ) u = 0 : ( A − r I ) u = ( [ 1 3 − 1 5 ] − 2 [ 1 0 0 1 ] ) [ u 1 u 2 ] = ( [ 1 3 − 1 5 ] − [ 2 0 0 2 ] ) [ u 1 u 2 ] = [ − 1 3 − 1 3 ] [ u 1 u 2 ] The matrix equation is: [ − 1 3 − 1 3 ] [ u 1 u 2 ] = [ 0 0 ] It can be write in scalar equation as, − u 1 + 3 u 2 = 0

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.5, Problem 43E

To determine

To find:

The general solution for the system tx′(t)=[13−15]x(t), t>0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.5, Problem 42E

Chapter 9.5, Problem 44E

Students have asked these similar questions

1.2.16. Let e be an edge appearing an odd number of times in a closed walk W. Prove that W contains the edges of a cycle through c.

1.2.11. (−) Prove or disprove: If G is an Eulerian graph with edges e, f that share vertex, then G has an Eulerian circuit in which e, f appear consecutively. a

By forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1

Chapter 9 Solutions

MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.

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 system t x ′ ( t ) = [ 1 3 − 1 5 ] x ( t ) , t > 0 .

Solution Summary: The author explains how to find the general solution for the given system by using the characteristics equation left|A-rIright|=0.

Videos

To find:

The general solution for the system tx′(t)=[13−15]x(t), t>0.

Want to see the full answer?

Chapter 9 Solutions

The general solution for the system t x ′ ( t ) = [ 1 3 − 1 5 ] x ( t ) , t &gt; 0 .

Solution Summary: The author explains how to find the general solution for the given system by using the characteristics equation left|A-rIright|=0.

Videos

To find: The general solution for the system tx′(t)=[13−15]x(t), t>0.

Want to see the full answer?

Chapter 9 Solutions

The general solution for the system t x ′ ( t ) = [ 1 3 − 1 5 ] x ( t ) , t > 0 .

To find:

The general solution for the system tx′(t)=[13−15]x(t), t>0.