Chapter 7 : Higher Order Differential Equations

In this chapter we’re going to take a look at higher order differential equations. This chapter will actually contain more than most text books tend to have when they discuss higher order differential equations.

We will definitely cover the same material that most text books do here. However, in all the previous chapters all of our examples were 2^nd order differential equations or \(2 \times 2\) systems of differential equations. So, in this chapter we’re also going to do a couple of examples here dealing with 3^rd order or higher differential equations with Laplace transforms and series as well as a discussion of some larger systems of differential equations.

Here is a brief listing of the topics in this chapter.

Basic Concepts for \(n^{\text{th}}\) Order Linear Equations – In this section we’ll start the chapter off with a quick look at some of the basic ideas behind solving higher order linear differential equations. Included will be updated definitions/facts for the Principle of Superposition, linearly independent functions and the Wronskian.

Linear Homogeneous Differential Equations – In this section we will extend the ideas behind solving 2^nd order, linear, homogeneous differential equations to higher order. As we’ll most of the process is identical with a few natural extensions to repeated real roots that occur more than twice. We will also need to discuss how to deal with repeated complex roots, which are now a possibility. In addition, we will see that the main difficulty in the higher order cases is simply finding all the roots of the characteristic polynomial.

Undetermined Coefficients – In this section we work a quick example to illustrate that using undetermined coefficients on higher order differential equations is no different that when we used it on 2^nd order differential equations with only one small natural extension.

Variation of Parameters – In this section we will give a detailed discussion of the process for using variation of parameters for higher order differential equations. We will also develop a formula that can be used in these cases. We will also see that the work involved in using variation of parameters on higher order differential equations can be quite involved on occasion.

Laplace Transforms – In this section we will work a quick example using Laplace transforms to solve a differential equation on a 3^rd order differential equation just to say that we looked at one with order higher than 2^nd. As we’ll see, outside of needing a formula for the Laplace transform of \(y'''\), which we can get from the general formula, there is no real difference in how Laplace transforms are used for higher order differential equations.

Systems of Differential Equations – In this section we’ll take a quick look at extending the ideas we discussed for solving \(2 \times 2\) systems of differential equations to systems of size \(3 \times 3\). As we will see they are mostly just natural extensions of what we already know how to do. We will also make a couple of quick comments about \(4 \times 4\) systems.

Series Solutions – In this section we are going to work a quick example illustrating that the process of finding series solutions for higher order differential equations is pretty much the same as that used on 2^nd order differential equations.