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.

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 .