2 - First-Order Ordinary Differential Equations

Abstract

This chapter focuses on frequently used first-order ordinary differential equations and methods to construct their solutions. Differential equations are first encountered in the beginning integral calculus courses. Although the phrase differential equation is not frequently used at that point, the problem of finding a function whose derivative is a given function is a differential equation. Some first-order equations can be classified as separable equations, others as homogeneous, and others as exact equations. However, most differential equations are neither separable, nor homogeneous, nor exact. A differential equation that can be written in the form, g(y)y1 = m, is called a separable differential equation. Separable differential equations are solved by collecting all the terms involving y on one side of the equation and all the terms involving x on the other side of the equation and integrating. Therefore, in the case of a separable differential equation, the variables and integrate are separated on both sides of the equation. As any differential equation can be accompanied by one or more auxiliary conditions, a separable equation can be stated along with an initial condition.