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Chapter 3 : Second Order Differential Equations

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 – In this section give an in depth discussion on the process used to solve homogeneous, linear, second order differential equations, \(ay'' + by' + cy = 0\). We derive the characteristic polynomial and discuss how the Principle of Superposition is used to get the general solution.

Real Roots – In this section we discuss the solution to homogeneous, linear, second order differential equations, \(ay'' + by' + c = 0\), in which the roots of the characteristic polynomial, \(ar^{2} + br + c = 0\), are real distinct roots.

Complex Roots – In this section we discuss the solution to homogeneous, linear, second order differential equations, \(ay'' + by' + c = 0\), in which the roots of the characteristic polynomial, \(ar^{2} + br + c = 0\), are complex roots. We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers.

Repeated Roots – In this section we discuss the solution to homogeneous, linear, second order differential equations, \(ay'' + by' + c = 0\), in which the roots of the characteristic polynomial, \(ar^{2} + br + c = 0\), are repeated, i.e. double, roots. We will use reduction of order to derive the second solution needed to get a general solution in this case.

Reduction of Order – In this section we will take 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 – In this section we will a look at some of the theory behind the solution to second order differential equations. We define fundamental sets of solutions and discuss how they can be used to get a general solution to a homogeneous second order differential equation. We will also define the Wronskian and show how it can be used to determine if a pair of solutions are a fundamental set of solutions.

More on the Wronskian – In this section we will examine how the Wronskian, introduced in the previous section, can be used to determine if two functions are linearly independent or linearly dependent. We will also give and an alternate method for finding the Wronskian.

Nonhomogeneous Differential Equations – In this section we will discuss the basics of solving nonhomogeneous differential equations. We define the complimentary and particular solution and give the form of the general solution to a nonhomogeneous differential equation.

Undetermined Coefficients – In this section we introduce the method of undetermined coefficients to find particular solutions to nonhomogeneous differential equation. We work a wide variety of examples illustrating the many guidelines for making the initial guess of the form of the particular solution that is needed for the method.

Variation of Parameters – In this section we introduce the method of variation of parameters to find particular solutions to nonhomogeneous differential equation. We give a detailed examination of the method as well as derive a formula that can be used to find particular solutions.

Mechanical Vibrations – In this section we will examine mechanical vibrations. In particular we will model an object connected to a spring and moving up and down. We also allow for the introduction of a damper to the system and for general external forces to act on the object. Note as well that while we example mechanical vibrations in this section a simple change of notation (and corresponding change in what the quantities represent) can move this into almost any other engineering field.