Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra With Applications
Found in: Page 441
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Find the real solution of the system dxdt=[04-90]x

The solution is xt=2sin6t2cos6t3cos6t-3sin6txy

See the step by step solution

Step by Step Solution

Step 1: Definition of the theorem

Continuous dynamical system with eigenvalues

Consider the linear system dxdt=Axwhere A is a real 2×2 matrix with complex eigenvalues p±iq (and q0 ).

Consider an eigenvector role="math" localid="1659877431239" v+iwwith eigenvalue p±iq.

Then x(t)=eptS[cosqt-sinqtsinqtcosqt]S-1x0 where S=[wv]. Recall that S-1x0 is the coordinate vector of x0 with respect to basis w,v.

Step 2: To find the eigenvalues

Consider the given system as follows.


No, to find the eigenvalues of the coefficient matrix as follows.


Simplify further as follows


Therefore, the eigenvalues are λ1=6i and λ2=-6i

Step3: To find the eigenvectors

To find a formula for trajectory as follows.


Therefore, the eigenvectors are as follows.


Then by the theorem, the general solution for the system dxdt=04-90xis as follows.

xt=eptScosqt-sinqtsinqtcosqtS-1x0 where S=wv

Similarly, the values of p, q and S are as follows.


Substitute the value 0 for p and 6 for q and 1001 for S in xt=eptScosqt-sinqtsinqtcosqtS-1x0 as follows.


Hence, the solution for the system dxdt=04-90xis xt=2sin6t2cos6t3cos6t-3sin6txy

where x and y are arbitrary constants.

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.