ABSTRACT
In previous chapters, it was seen how separable and linear differential equations can be solved using methods for converting them to forms that can be easily integrated. This chapter develops a more a more general approach to converting a differential equation to a form (the “exact form”) that can be integrated through a relatively straightforward procedure. It will be seen what it means for an equation to be in exact form and how to solve equations in this form. Because it is not always obvious when a given equation is in exact form, a practical “test for exactness” is also be developed. Finally, the notion of integrating factors is generalized to help find exact forms for a variety of first-order differential equations.
The theory and methods developed in this chapter are more general than those developed earlier for separable and linear equations. In fact, the procedures developed here can be used to solve any separable or linear differential equation (though most would probably prefer using the methods developed earlier). More importantly, the methods developed in this chapter can, in theory at least, be used to solve a great number of other first-order differential equations.
