In this section we want to take a brief look at systems of
differential equations that are larger than 2 x 2. The problem here is that unlike the first few
sections where we looked at n^{th}
order differential equations we can’t really come up with a set of formulas
that will always work for every system.
So, with that in mind we’re going to look at all possible cases for a 3
x 3 system (leaving some details for you to verify at times) and then a couple
of quick comments about 4 x 4 systems to illustrate how to extend things out to
even larger systems and then we’ll leave it to you to actually extend things
out if you’d like to.
We will also not be doing any actual examples in this
section. The point of this section is
just to show how to extend out what we know about 2 x 2 systems to larger
systems.
Initially the process is identical regardless of the size of
the system. So, for a system of 3
differential equations with 3 unknown functions we first put the system into
matrix form,
where the coefficient matrix, A, is a 3 x 3 matrix. We
next need to determine the eigenvalues and eigenvectors for A and because A is a 3 x 3 matrix we know that there will be 3 eigenvalues
(including repeated eigenvalues if there are any).
This is where the process from the 2 x 2 systems starts to
vary. We will need a total of 3 linearly
independent solutions to form the general solution. Some of what we know from the 2 x 2 systems
can be brought forward to this point.
For instance, we know that solutions corresponding to simple eigenvalues
(i.e. they only occur once in the
list of eigenvalues) will be linearly independent. We know that solutions from a set of complex
conjugate eigenvalues will be linearly independent. We also know how to get a set of linearly
independent solutions from a double eigenvalue with a single eigenvector.
There are also a couple of facts about eigenvalues/eigenvectors
that we need to review here as well.
First, provided A has only
real entries (which it always will here) all complex eigenvalues will occur in
conjugate pairs (i.e. ) and their associated eigenvectors
will also be complex conjugates of each other.
Next, if an eigenvalue has multiplicity (i.e.
occurs at least twice in the list of eigenvalues) then there will be anywhere
from 1 to k linearly independent
eigenvectors for the eigenvalue.
With all these ideas in mind let’s start going through all
the possible combinations of eigenvalues that we can possibly have for a 3 x 3
case. Let’s also note that for a 3 x 3 system
it is impossible to have only 2 real distinct eigenvalues. The only possibilities are to have 1 or 3
real distinct eigenvalues.
Here are all the possible cases.
3 Real Distinct
Eigenvalues
In this case we’ll have the real, distinct eigenvalues and their associated eigenvectors, ,
and are guaranteed to be linearly independent and
so the three linearly independent solutions we get from this case are,
1 Real and 2 Complex
Eigenvalues
From the real eigenvalue/eigenvector pair, and ,
we get one solution,
We get the other two solutions in the same manner that we
did with the 2 x 2 case. If the eigenvalues are with eigenvectors and we can get two real-valued solution by using Euler’s formula to expand,
into its real and imaginary parts, . The final two real valued solutions we need
are then,
1 Real Distinct and 1
Double Eigenvalue with 1 Eigenvector
From the real eigenvalue/eigenvector pair, and ,
we get one solution,
From our work in the 2 x 2 systems we know that from the double eigenvalue with single eigenvector, ,
we get the following two solutions,
where and must satisfy the following equations,
Note that the first equation simply tells us that must be the single eigenvector for this
eigenvalue, ,
and we usually just say that the second solution we get from the double root
case is,
1 Real Distinct and 1
Double Eigenvalue with 2 Linearly Independent Eigenvectors
We didn’t look at this case back when we were examining the
2 x 2 systems but it is easy enough to deal with. In this case we’ll have a single real
distinct eigenvalue/eigenvector pair, and ,
as well as a double eigenvalue and the double eigenvalue has two linearly
independent eigenvectors, and .
In this case all three eigenvectors are linearly independent
and so we get the following three linearly independent solutions,
We are now out of the cases that compare to those that we
did with 2 x 2 systems and we now need to move into the brand new case that we
pick up for 3 x 3 systems. This new case
involves eigenvalues with multiplicity of 3.
As we noted above we can have 1, 2, or 3 linearly independent
eigenvectors and so we actually have 3 sub cases to deal with here. So let’s go through these final 3 cases for a
3 x 3 system.
1 Triple Eigenvalue
with 1 Eigenvector
The eigenvalue/eigenvector pair in this case are and . Because the eigenvalue is real we know that
the first solution we need is,
We can use the work from the double eigenvalue with one
eigenvector to get that a second solution is,
For a third solution we can take a clue from how we dealt
with n^{th} order
differential equations with roots multiplicity 3. In those cases we multiplied the original
solution by a . However, just as with the double eigenvalue
case that won’t be enough to get us a solution.
In this case the third solution will be,
where ,
,
and must satisfy,
You can verify that this is a solution and the conditions by
taking a derivative and plugging into the system.
Now, the first condition simply tells us that because we only have a single eigenvector here
and so we can reduce this third solution to,
where ,
and must satisfy,
and finally notice that we would have solved the new first
condition in determining the second solution above and so all we really
need to do here is solve the final condition.
As a final note in this case, the is in the solution solely to keep any extra
constants from appearing in the conditions which in turn allows us to reuse
previous results.
1 Triple Eigenvalue
with 2 Linearly Independent Eigenvectors
In this case we’ll have the eigenvalue with the two linearly independent eigenvectors
and so we get the following two linearly
independent solutions,
We now need a third solution. The third solution will be in the form,
where and must satisfy the following equations,
We’ve already verified that this will be a solution with
these conditions in the double eigenvalue case (that work only required a
repeated eigenvalue, not necessarily a double one).
However, unlike the previous times we’ve seen this we can’t
just say that is an eigenvector. In all the previous cases in which we’ve seen
this condition we had a single eigenvalue and this time we have two linearly
independent eigenvalues. This means that
the most general possible solution to the first condition is,
This creates problems in solving the second condition. The second condition will not have solutions
for every choice of and and the choice that we use will be dependent
upon the eigenvectors. So upon solving
the first condition we would need to plug the general solution into the second
condition and then proceed to determine conditions on and that would allow us to solve the second
condition.
1 Triple Eigenvalue
with 3 Linearly Independent Eigenvectors
In this case we’ll have the eigenvalue with the three linearly independent
eigenvectors ,
,
and so we get the following three linearly
independent solutions,
4 x 4 Systems
We’ll close this section out with a couple of comments about
4 x 4 systems. In these cases we will
have 4 eigenvalues and will need 4 linearly independent solutions in order to
get a general solution. The vast
majority of the cases here are natural extensions of what 3 x 3 systems cases
and in fact will use a vast majority of that work.
Here are a couple of new cases that we should comment
briefly on however. With 4 x 4 systems
it will now be possible to have two different sets of double eigenvalues and
two different sets of complex conjugate eigenvalues. In either of these cases we can treat each
one as a separate case and use our previous knowledge about double eigenvalues
and complex eigenvalues to get the solutions we need.
It is also now possible to have a “double” complex
eigenvalue. In other words we can have each occur twice in the list of
eigenvalues. The solutions for this case
aren’t too bad. We get two solutions in
the normal way of dealing with complex eigenvalues. The remaining two solutions will come from
the work we did for a double eigenvalue.
The work we did in that case did not require that the
eigenvalue/eigenvector pair to be real.
Therefore if the eigenvector associated with is then the second solution will be,
and once we’ve determined we can again split this up into its real and
imaginary parts using Euler’s formula to get two new real valued solutions.
Finally with 4 x 4 systems we can now have eigenvalues with
multiplicity of 4. In these cases we can
have 1, 2, 3, or 4 linearly independent eigenvectors and we can use our work
with 3 x 3 systems to see how to generate solutions for these cases. The one issue that you’ll need to pay
attention to is the conditions for the 2 and 3 eigenvector cases will have the
same complications that the 2 eigenvector case has in the 3 x 3 systems.
So, we’ve discussed some of the issues involved in systems
larger than 2 x 2 and it is hopefully clear that when we move into larger
systems the work can be become vastly more complicated.