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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

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Short Answer

Find the real solution of the system dxdt=[0-330]x

The solution is xt=cos3t-sin3tsin3tcos3txy

See the step by step solution

Step by Step Solution

Step1: definition of the theorem.

Continuous dynamical system with eigenvalues

Consider the linear system dxdt=Axwhere A is a real 2×2matrix with complex eigenvalues role="math" localid="1659881702011" p±iq (and q0).

Consider an eigenvector v+iw with eigenvalue p±iq.

Then x(t)=eptS[cosqt-sinqtsinqtcosqt]S-1x0where S=[wv]. Recall that S-1x0is the coordinate vector of x0with respect to basis w,v.

Step2: 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 λ=3i and λ=-3i

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=0-330xis as follows.


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



Substitute the value 0 for p and 3 for q and 1001 for S in xt=eptScosqt-sinqtsinqtcosqtS-1x0as follows.


Hence, the solution for the system dxdt=0-330xis xt=cos3t-sin3tsin3tcos3txy

where x and y are arbitrary constants.

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