TABLE OF CONTENTS We will now turn to second order differential equations,
In this chapter we address the kinds of theoretical question that we covered for first order equations in Chapter 6. As such there are few examples, but we will return to more concrete problems and solution methods in the next chapter.
First we discuss the existence and uniqueness of solutions. Before we do this formally, we give an indication of why we will need to specify both x and 
in our initial condition .
Consider the simplest type of second order equation,
the second order equivalent of the trivial equations we considered in Chapter 5. We can solve (11.2) by integrating twice: if F is any anti-derivative of f then
and then if 
is any anti-derivative of F
The two integrations result in two arbitrary constants ( c 1 and c 2); specifying x( t 0) alone will not be enough to tie down the solution, but we need to specify ( t 0) (to fix c 1) and then x( t 0) (to determine c 2).
To put this in a physical context, equation (11.2) is the equation of motion for a...
