Example 1 Solve
the following IVP.
Solution
We first need the eigenvalues and eigenvectors for the
matrix.
So, now that we have the eigenvalues recall that we only
need to get the eigenvector for one of the eigenvalues since we can get the
second eigenvector for free from the first eigenvector.
:
We need to solve the following system.
Using the first equation we get,
So, the first eigenvector is,
When finding the eigenvectors in these cases make sure
that the complex number appears in the numerator of any fractions since we’ll
need it in the numerator later on.
Also try to clear out any fractions by appropriately picking the
constant. This will make our life
easier down the road.
Now, the second eigenvector is,
However, as we will see we won’t need this eigenvector.
The solution that we get from the first eigenvalue and
eigenvector is,
So, as we can see there are complex numbers in both the
exponential and vector that we will need to get rid of in order to use this
as a solution. Recall from the complex
roots section of the second order differential equation chapter that we can
use Euler’s formula to get the
complex number out of the exponential.
Doing this gives us,
The next step is to multiply the cosines and sines into
the vector.
Now combine the terms with an “i” in them and split these terms off from those terms that don’t
contain an “i”. Also factor the “i” out of this vector.
Now, it can be shown (we’ll leave the details to you) that
and are two linearly independent solutions to
the system of differential equations.
This means that we can use them to form a general solution and they
are both real solutions.
So, the general solution to a system with complex roots is
where and are found by writing the first solution as
For our system then, the general solution is,
We now need to apply the initial condition to this to find
the constants.
This leads to the following system of equations to be
solved,
The actual solution is then,
