## ABSTRACT

This chapter provides the transition from first-order differential equations to differential equations of order two and higher. For the most part, the methods that were developed for solving first-order equationsare of limited direct use in solving higher-order equations, though some of the ideas have more subtle applications. That said, some second- (and a few higher-) order equations can be converted to first-order equations through clever substitutions, solved as such and the results then integrated to obtain the full solutions. That is the first topic in this chapter. This chapter also extends many of the concepts originally made with first-order equations to higher-order equations. These include “autonomous equations”, “general solutions”, “initial-value problems” and the “derivative formula form for a differential equation”. Finally, theorems on the existence and uniqueness of solutions to second- and higher-order initial-value problems are discussed. These theorems are analogs of the corresponding theorems for first-order initial-value problems given in a previous chapter.