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# Solution: In Exercises 9–18, construct the general solution

ISBN: 9780321982384 49

## Solution for problem 11E Chapter 5.7

Linear Algebra and Its Applications | 5th Edition

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Problem 11E

In Exercises 9–18, construct the general solution of  involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.

Step-by-Step Solution:

Solution  11E

Step 1 of 7</p>

Consider the matrix differential equation:

, where.

The objective is to find the general solution of the differential equation.

Find the eigenvalues of the matrix :

The characteristic polynomial is,

The roots of the characteristic equation are, .

Step 2 of 7</p>

Find the Eigen vectors of  for the corresponding Eigen values .

Let  be the Eigen vector of the matrix for the Eigen value .

Thus, the obtained equations are .

Step 3 of 7</p>

Multiply the first equation  by .

Thus, the obtained two equations represent the same equation. That is,

Thus, the Eigen vector is,

Take , then the Eigen vector is  for the Eigen value .

Step 4 of 7</p>

Let  be the Eigen vector of the matrix for the Eigen value .

Thus, the obtained equations are .

Step 5 of 7

Step 6 of 7

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