Example. Consider the system
Answer. The matrix coefficient of the system is
In order to find the eigenvalues consider the characteristic polynomial
Since , we have a repeated eigenvalue equal to 3. Let us find the associated eigenvector . Set
Then we must have which translates into
This reduces to y=x. Hence we may take
Next we look for the second vector . The equation giving this vector is which translates into the algebraic system
where
Clearly the two equations reduce to the equation y - x=1 or y = 1 + x, where x may be chosen to be any number. So if we take x=0 for example, we get
Therefore the two independent solutions are
The general solution will then be
In order to find the solution which satisfies the initial condition
we must have
This implies and . Hence the solution is
The phase plane with some solutions is given in the picture below:
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Author: Mohamed Amine Khamsi