Page 38 - Schaum's Outline of Differential Equations

CHAPTER 4

Separable First-Order

Differential Equations

GENERAL SOLUTION

The solulion lo Ihe first-order separable differential equation (see Chapter 3)

is

where c represents an arbitrary constant.
The integrals obtained in Hq. (4.2) may be. for all practical purposes, impossible lo evaluate. In such eases.
numerical techniques (see Chapters 18, 14. 20) are used to obtain an approximate solution. Even if the indicated
integrations in (4.2) can be performed, il may not be algebraical!} possible lo solve for y explicitly in terms of
x. In that case, the solution is left in implicit form.

SOLUTIONS TO THE INITIAL- VALUE PROBLEM
The solulion to the initial-value problem

can be obtained, as usual. b\ first using Hq. (4.2) to solve the differential equation and then applying Ihe initial
condition directly lo evaluate r.
Alternatively, the solulion lo hq. (4.3) can be obtained from

Liquation (4.4). however, may not determine the solulion of (43) uniquely: lhal is. (4.4) may have many solutions.
w
of which only one ill satisfy Ihe initial-value problem.
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