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.