Systems of Differential
Equations
In the introduction to this section we briefly discussed how
a system of differential equations can arise from a population problem in which
we keep track of the population of both the prey and the predator. It makes sense that the number of prey
present will affect the number of the predator present. Likewise, the number of predator present will
affect the number of prey present.
Therefore the differential equation that governs the population of
either the prey or the predator should in some way depend on the population of
the other. This will lead to two
differential equations that must be solved simultaneously in order to determine
the population of the prey and the predator.
The whole point of this is to notice that systems of
differential equations can arise quite easily from naturally occurring
situations. Developing an effective
predatorprey system of differential equations is not the subject of this
chapter. However, systems can arise from
n^{th} order linear
differential equations as well. Before
we get into this however, let’s write down a system and get some terminology
out of the way.
We are going to be looking at first order, linear systems of
differential equations. These terms mean
the same thing that they have meant up to this point. The largest derivative anywhere in the system
will be a first derivative and all unknown functions and their derivatives will
only occur to the first power and will not be multiplied by other unknown
functions. Here is an example of a
system of first order, linear differential equations.
We call this kind of system a coupled system since knowledge of x_{2} is required in order to find x_{1} and likewise knowledge of x_{1} is required to find x_{2}. We will worry
about how to go about solving these later. At this point we are only interested in
becoming familiar with some of the basics of systems.
Now, as mentioned earlier, we can write an n^{th} order linear differential
equation as a system. Let’s see how that
can be done.
Example 1 Write
the following 2^{nd} order
differential equation as a system of first order, linear differential
equations.
Solution
We can write higher order differential equations as a
system with a very simple change of variable.
We’ll start by defining the following two new functions.
Now notice that if we differentiate both sides of these we
get,
Note the use of the differential equation in the second
equation. We can also convert the
initial conditions over to the new functions.
Putting all of this together gives the following system of
differential equations.

We will call the system in the above example an Initial Value Problem just as we did
for differential equations with initial conditions.
Let’s take a look at another example.
Example 2 Write
the following 4^{th} order differential equation as a system of
first order, linear differential equations.
Solution
Just as we did in the last example we’ll need to define
some new functions. This time we’ll
need 4 new functions.
The system along with the initial conditions is then,

Now, when we finally get around to solving these we will see
that we generally don’t solve systems in the form that we’ve given them in this
section. Systems of differential
equations can be converted to matrix
form and this is the form that we usually use in solving systems.
Example 3 Convert the following system to matrix from.
Solution
First write the system so that each side is a vector.
Now the right side can be written as a matrix
multiplication,
Now, if we define,
then,
The system can then be written in the matrix form,

Example 4 Convert
the systems from Examples 1 and 2 into matrix form.
Solution
We’ll start with the system from Example 1.
First define,
The system is then,
Now, let’s do the system from Example 2.
In this case we need to be careful with the t^{2} in the last
equation. We’ll start by writing the
system as a vector again and then break it up into two vectors, one vector
that contains the unknown functions and the other that contains any known
functions.
Now, the first vector can now be written as a matrix
multiplication and we’ll leave the second vector alone.
where,
Note that occasionally for “large” systems such as this we
will go one step farther and write the system as,

The last thing that we need to do in this section is get a
bit of terminology out of the way.
Starting with
we say that the system is homogeneous if and we say the system is nonhomogeneous if .