digit 135

2

dy

–

– + f(x)y= g(x)

dx

dy y

–

– +

–

= x2

dx x

1C

y=

–

x3+

–

4x

1

xy =

–

x4+ C

4

d

–

– (xy) = x3

dx

dy

––

dx

Homogeneous Equations: A first-order equation of the

form:

can be solved by substituting y = vx. This reduces the

equation to one in vand xalone. Separation of vari-

ables will now work.

For example, to solve , which can

be written , set y= vx to obtain:

Separation of variables now applies.

Linear Equations: A first-order equation of the form:

can be solved by multiplying through by the integrating

factor e∫f(x)dx. This yields the equation:

which can be rewritten:

This is directly integrable.

For example, the equation has inte-

grating factor , and so the equation

can be rewritten , yielding ,

or .

Basic Second-Order Equations: A second-order equa-

tion of the form:

can be solved by multiplying through by 2 to

obtain:

Integrating gives:

After taking square roots, one can now separate

variables.

It is often the case that no known techniques will

solve a particular differential equation that arises in a

particular scientific study. In this case, mathematicians

will often assume that the solution function y= f(x) can

be written as a T

AYLOR SERIES

: y= a0+ a1x+ a2x2

+a3x3+ … By substituting the series into the differen-

tial equation, it is usually possible to compute the val-

ues of at least the first few coefficients a0,a1,a2,…. This

approach is called the “method of undetermined coeffi-

cients.” In this context, the Taylor series used is some-

times called a perturbation function.

digit A symbol that forms part of a number is called

a digit. For example, the number 42.768 has five digits.

Ten digits are used in decimal notation, namely, 0, 1, 2,

3, 4, 5, 6, 7, 8, and 9. In hexadecimal notation

(base 16), the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,

C, D, E, and F. In the system of binary numbers only

two digits are used: 0 and 1. In general, a counting sys-

tem in base buses bdifferent digits to represent the

numbers 0 through b– 1.

In the vernacular, the word digit means a finger or

a toe. As one learns to count with one’s digits it is not

surprising that the word has come to be used for spe-

cific numbers the fingers represent. The measure of a

digit, defined as the width of one finger, about 3/4of an

inch, was a standard Old English unit of

LENGTH

.

A digital watch displays specific symbols (num-

bers) as discrete units of time (seconds) pass, and a

digital computer processes information supplied to it

in the form of numbers. An analog watch, however,

uses the sweeping entity of a moving hand as an ana-

log to the passing of time, and an analog computer, now

generally considered obsolete, uses a varying voltage to