Page 31 - Schaum's Outline of Differential Equations

CHAPTER 3

Classifications

of First-Order

Differential Equations

STANDARD FORM AND DIFFERENTIAL FORM

Standard form fora lirsl-order differential equation in ihe unknown I'unciion y(x) is

where the derivative y'appears onl\ on the left side of UJ). Many, hill not all. first-order differential equations
can be written in standard form b> algebraical!} solving for y' and then setting/Or, y) equal to ihe right side of
Ihe resulting equation.
The right side of (J./Jcan always be written as a quotient of two other functions M(.r, v) and-M.v.y). Then
(3.}) becomes dy/dx = M(x. y)/-N(x. y). which is equivalent to Ihe differential farm

LINEAR EQUATIONS
Consider a differential equation in standard form (3.1}. If./i.v, y| can be written as/(jr. y) = —p(x)y + q(x)
(that is, as a function of .v times y. plus another function of .v). ihe differential equaiion is linear. First-order
linear differential equations can alwavs be e\pressed as

Linear equations are solved in Chapter 6.

BERNOULLI EQUATIONS
A Bernoulli differential equaiion is an equation of the form

where n denotes a real number. When n = 1 or n = 0. a Bernoulli equaiion reduces to a linear equaiion. Bernoulli
equations are solved in Chapter 6.

14
Copyright © 2006, 1994, 1973 by The McGraw-Hill Companies, Inc. Click here for terms of use.