Example: Consider the linear system

.

Find the matrix coefficient of the system. Then, discuss the fate of the long term behavior of the solutions. If they go to infinity, discuss how.

Answer: The matrix coefficient of the system is

.

Note that if you have the wrong matrix coefficient the conclusion about the solutions may totally differ from the right answer!
In order to find the general solution we need the characteristic equation

.

Its roots are given by the quadratic formulas

,

which gives or . Next, we need to find the associated eigenvectors.

Case . Denote by the associated eigenvector. The system giving is

.

The two equations lead to the same equation . If we choose , we get

.

Case . Denote by , the associated eigenvector. The system giving is

The second equation is worthless and the first one implies . If we choose , we get

.

Therefore, the general solution is given by

,

where and are two parameters.

We know that since the system has one positive eigenvalue the solutions will tend to infinity as t goes to . We also know that the solutions will get closer and closer to the straight-line solution which corresponds to the biggest eigenvalues. In this case, the line generated by the vector

,

is clearly the x-axis. See the graph below.

[Differential Equations] [First Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi