Second Order Differential Equations

Introduction

In the previous chapter we looked at first order differential equations. In this chapter we will move on to second order differential equations. Just as we did in the last chapter we will look at some special cases of second order differential equations that we can solve. Unlike the previous chapter however, we are going to have to be even more restrictive as to the kinds of differential equations that we’ll look at. This will be required in order for us to actually be able to solve them.

Here is a list of topics that will be covered in this chapter. Basic Concepts – Some of the basic concepts and ideas that are involved in solving second order differential equations.

Real Roots – Solving differential equations whose characteristic equation has real roots. Complex Roots – Solving differential equations whose characteristic equation complex real roots.

Repeated Roots – Solving differential equations whose characteristic equation has repeated roots.

Reduction of Order – A brief look at the topic of reduction of order. This will be one of the few times in this chapter that non-constant coefficient differential equation will be looked at.

Fundamental Sets of Solutions – A look at some of the theory behind the solution to second order differential equations, including looks at the Wronskian and fundamental sets of solutions.

More on the Wronskian – An application of the Wronskian and an alternate method for finding it.

Nonhomogeneous Differential Equations – A quick look into how to solve nonhomogeneous differential equations in general.

Undetermined Coefficients – The first method for solving nonhomogeneous differential equations that we’ll be looking at in this section.

Variation of Parameters – Another method for solving nonhomogeneous differential equations.

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Differential Equations

Mechanical Vibrations – An application of second order differential equations. This section focuses on mechanical vibrations, yet a simple change of notation can move this into almost any other engineering field.

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Differential Equations

Basic Concepts

In this chapter we will be looking exclusively at linear second order differential equations. The most general linear second order differential equation is in the form.

p ( t ) y′′ + q ( t ) y ′ + r ( t ) y = g ( t )

(1)

In fact, we will rarely look at non-constant coefficient linear second order differential equations. In the case where we assume constant coefficients we will use the following differential equation.

ay′′ + by ′ + cy = g ( t )

(2)

Where possible we will use (1) just to make the point that certain facts, theorems, properties, and/or techniques can be used with the non-constant form. However, most of the time we will be using (2) as it can be fairly difficult to solve second order non-constant coefficient differential equations.

Initially we will make our life easier by looking at differential equations with g(t) = 0. When g(t) = 0 we call the differential equation homogeneous and when g ( t ) ≠ 0 we call the differential equation nonhomogeneous.

So, let’s start thinking about how to go about solving a constant coefficient, homogeneous, linear, second order differential equation. Here is the general constant coefficient, homogeneous, linear, second order differential equation.

ay′′ + by′ + cy = 0

It’s probably best to start off with an example. This example will lead us to a very important fact that we will use in every problem from this point on. The example will also give us clues into how to go about solving these in general.

Example 1 Determine some solutions to

y ′′ − 9 y = 0

Solution

We can get some solutions here simply by inspection. We need functions whose second derivative is 9 times the original...