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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

##### ISBN: 9780321982384

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