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TERMINOLOGY AND SEPARABLE EQUATIONS
LINEAR EQUATIONS EXACT EQUATIONS
CHAPTER 1 HOMOGENEOUS BERNOULLI AND RICCATI
EQUATIONS EXISTENCE AND UNIQUENESS
First-Order

Differential

Equations

1.1 Terminology and Separable Equations

Part 1 of this book deals with ordinary differential equations, which are equations that contain
one or more derivatives of a function of a single variable. Such equations can be used to model a
rich variety of phenomena of interest in the sciences, engineering, economics, ecological studies,
and other areas.
We begin in this chapter with first-order differential equations, in which only the first
derivative of the unknown function appears. As an example,

y + xy = 0

is a first-order equation for the unknown function y(x).A solution of a differential equation is
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any function satisfying the equation. It is routine to check by substitution that y = ce −x /2 is a
solution of y + xy = 0 for any constant c.

We will develop techniques for solving several kinds of first-order equations which arise in
important contexts, beginning with separable equations.

A differential equation is separable if it can be written (perhaps after some algebraic
manipulation) as
dy
= F(x)G(y)
dx
in which the derivative equals a product of a function just of x and a function just of y.
This suggests a method of solution.

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