Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

First Order Differential Equations

his chapter deals with differential equations of first order,

where f is a given function of two variables. Any differentiable function y = (t) that satisfies this equation for all t in some interval is called a solution. Our object is to develop methods for finding solutions or, if that is not possible, approximating them. Unfortunately, for an arbitrary function f, there is no general method for solving Eq. (1) in terms of elementary functions. Instead, we will describe several methods, each of which is applicable to a certain subclass of first order equations. The most important of these are linear equations (already discussed in Section 1.2), separable equations (Section 2.1), and exact equations (Section 2.5). In Section 2.4, we discuss another subclass of first order equations, autonomous equations, for which geometrical methods yield valuable information about solutions. Finally, in Sections 2.6 and 2.7, we revisit the question of constructing numerical approximations to solutions, and introduce algorithms more efficient than the Euler method ...