## ABSTRACT

“Linear” equations make up an important class of differential equations that commonly arise in applications and which (in theory) are relatively easy to solve. As with the notion of ‘separability’, the idear of linearity for first-order differential equations can be viewed as a simple generalization of direct integability, and a relatively straightforward method can be used to put any first-order linear differential equation into a form that can be integrated.

In this chapter, the criteria for a first-order equation being “linear” are given, and a step-by-step method for solving these equations is derived. This method involves an “integrating factor”, a concept that will be expanded upon in later chapters. This method also leads to a concise, general formula for solving linear first-order equations, the advantages and disadvantages of which are discussed near the end of the chapter.

It should be noted that the criteria for first-order differential equations being linear will, in later chapters, be expanded to criteria for higher-order differential equations being linear, and that a rather extensive theory will be developed to deal with those higher-order linear equations. That theory is not needed for first-order linear equations; in fact, for first-order equations it is of very limited value.