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.
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.