Solution of system of second order ordinary differential equation
Answers
The whole point of this is to notice that systems of differential equations can arise quite easily from naturally occurring situations. Developing an effective predator-prey 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.
x
′
1
=
x
1
+
2
x
2
x
′
2
=
3
x
1
+
2
x
2
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.