1550078481-Ordinary_Differential_Equations__Roberts_

58 Ordinary Differential Equations

2.3.2 Solution of the Separable Equation y' = g(x)/h(y)

From your study of calculus, you are already familiar with the concept of
explicit and implicit functions and you know how to differentiate both explic-
itly and implicitly. In differential equations, we distinguish between explicit
and implicit solutions as well. Thus far in the text, we have encountered only
explicit solutions to differential equations; therefore, we have referred to them
simply as solutions. The definitions of explicit solution and implicit solution
for first-order differential equations follows.

DEFINITIONS Explicit Solution and Implicit Solution

An explicit solution of the differential equation y' = f(x, y) on an

interval I = (a, b) is a function y = ¢(x) which is differentiable at least

once on I and satisfies q/ = f(x, ¢(x)).

A relation R(x, y) = 0 is an implicit solution of the y' = f(x, y) on an

interval I if the relation defines at least one function y 1 ( x) on I such that

y 1 (x) is an explicit solution of y' = f(x, y) on I.

Usually we refer to both explicit and implicit solutions simply as solutions.
However, when we want to be specific about the kind of solution we are talking
about, we will include the designation explicit or implicit.

We so lved the radioactive differential equation (1) of section 1.4 by "sep-
arating" the variables. Differential equations which can be solved using this
technique are called separable equations and are defined as follows.

DEFINITION Separable Equation

The differential equation y' = f(x, y) is a separable equation, if it can

be written in the form

(9) y'

g(x)

h(y)'

In most cases a separable differential equation will not be in the form (9) ini-
tially. Some algebraic manipulation will usually be required in order to write
the given equation in the form (9). Of course, most differential equations
are not separable; and, therefore, no amount of valid algebraic manipulation
will produce an equation of the form (9). For example, the differential equa-

tion y' = (x - y)/x is not separable. When possible, the algebraic process

of converting y' = f(x, y) into the form y' = g(x)/h(y) is called "separating