We’ll start this chapter off with the material that most
text books will cover in this chapter.
We will take the material from the Second
Order chapter and expand it out to n^{th}
order linear differential equations. As
we’ll see almost all of the 2^{nd} order material will very naturally
extend out to n^{th} order
with only a little bit of new material.
So, let’s start things off here with some basic concepts for
n^{th} order linear differential
equations. The most general n^{th} order linear differential
equation is,


(1)

where you’ll hopefully recall that,
Many of the theorems and ideas for this material require
that has a coefficient of 1 and so if we divide out
by we get,
As we might suspect an IVP for an n^{th} order differential equation will require the
following n initial conditions.
The following theorem tells us when we can expect there to
be a unique solution to the IVP given by (2) and
(3).
Theorem 1
This theorem is a very natural extension of a similar
theorem we saw in the 1^{st} order
material.
Next we need to move into a discussion of the n^{th} order linear homogeneous
differential equation,
Let’s suppose that we know are all solutions to (4)
then by the an extension of the Principle
of Superposition we know that
will also be a solution to (4). The real question here is whether or not this
will form a general solution to (4).
In order for this to be a general solution then we will have
to be able to find constants for any choice of (in the interval I from Theorem 1) and any choice of . Or, in other words we need to be able to
find that will solve,
Just as we did
for 2^{nd} order differential equations, we can use Cramer’s Rule to solve this and the
denominator of each the answers will be the following determinant of an n x n
matrix.
As we did back with the 2^{nd}
order material we’ll define this to be the Wronskian
and denote it by,
Now that we have the definition of the Wronskian out of the
way we need to get back to the question at hand. Because the Wronskian is the denominator in the
solution to each of the we can see that we’ll have a solution provided
it is not zero for any value of that we chose to evaluate the Wronskian at. The following theorem summarizes all this up.
Theorem 2
Recall as well that if a set of solutions form a fundamental
set of solutions then they will also be a set of linearly
independent functions.
We’ll close this section off with a quick reminder of how we
find solutions to the nonhomogeneous differential equation, (2). We first need the n^{th} order version of a theorem we saw back in the 2^{nd} order material.
Theorem 3
Now, just as we did with the 2^{nd} order material
if we let be the general solution to (2)
and if we let be any solution to (2)
then using the result of this theorem we see that we must have,
where, is called the complementary solution and is called a particular solution.
Over the course of the next couple of sections we’ll discuss
the differences in finding the complementary and particular solutions for n^{th} order differential
equations in relation to what we know about 2^{nd} order differential
equations. We’ll see that, for the most
part, the methods are the same. The
amount of work involved however will often be significantly more.