Answer to Example 3

Example: Consider the linear system

displaymath60.

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

displaymath62.

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

displaymath64.

Its roots are given by the quadratic formulas

displaymath66,

which gives tex2html_wrap_inline68 or tex2html_wrap_inline70 . Next, we need to find the associated eigenvectors.

Case tex2html_wrap_inline68 . Denote by tex2html_wrap_inline74 the associated eigenvector. The system giving tex2html_wrap_inline76 is

displaymath78.

The two equations lead to the same equation tex2html_wrap_inline80 . If we choose tex2html_wrap_inline82 , we get

displaymath84.

Case tex2html_wrap_inline70 . Denote by tex2html_wrap_inline88, the associated eigenvector. The system giving tex2html_wrap_inline90 is

displaymath92

The second equation is worthless and the first one implies tex2html_wrap_inline94 . If we choose tex2html_wrap_inline82 , we get

displaymath98.

Therefore, the general solution is given by

displaymath100,

where tex2html_wrap_inline102 and tex2html_wrap_inline104 are two parameters.

We know that since the system has one positive eigenvalue the solutions will tend to infinity as t goes to tex2html_wrap_inline108 . 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

displaymath110,

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



[Differential Equations] [First Order D.E.]
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Author: Mohamed Amine Khamsi

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