Lecture (

25

):

Ordinary Differential Equations (

1 of 2

)

A

differential equation

is an algebraic equation

that contains some

derivatives

:

•

Recall that a

derivative

indicates a change in a

dependent variable

with respect to an

independent variable

.

•

In these two examples,

y

is the

dependent

variable

and

t

and

x

are the

independent

variables

, respectively.

Why study differential equations?

•

Many descriptions of natural phenomena are

relationships (

equations

) involving the

rates

at

which things happen (

derivatives

).

•

Equations containing

derivatives

are called

differential equations

.

•

Ergo, to investigate problems in many fields of

science and technology, we need to know

something about

differential equations

.

Why study differential equations?

•

Some examples of fields using differential

equations in their analysis include:

—

Solid mechanics & motion

—

heat transfer & energy balances

—

vibrational dynamics & seismology

—

aerodynamics & fluid dynamics

—

electronics & circuit design

—

population dynamics & biological systems

—

climatology and environmental analysis

—

options trading & economics

Examples of Fields Using Differential

Equations in Their Analysis

Differential Equation Basics

•

The order of the highest derivative in

a differential equation indicates the

order of

the equation

.

Simple Differential Equations

A

simple differential equation

has the

form

Its general solution is

Ex.

Find the general solution to

Simple Differential Equations

Ex.

Find the general solution to

Simple Differential Equations

Find the general solution to

Exercise:

(Waner, Problem #1, Section 7.6)

A drag racer accelerates from a stop so that

its speed is

40

t

feet per second

t

seconds after

starting. How

far

will the car go in

8

seconds?

Example:

Motion

Given:

Find:

Solution:

Apply the initial condition:

s

(0) = 0

The car travels 1280 feet in 8 seconds

Find the particular solution to

Exercise:

(Waner, Problem #11, Section 7.6)

Apply the initial condition:

y

(0) = 1

Separable

Differential Equations

A

separable differential equation

has the form

Its general solution is

Consider the differential equation

Example:

Separable Differential Equation

a. Find the general solution.

b. Find the particular solution that satisfies

the initial condition

y

(0) = 2

.

Solution:

Step

1

—

Separate the variables:

Step

2

—

Integrate both sides:

Step

3

—

Solve for the dependent variable:

a

.

This is the

general

solution

Solution:

(continued)

Apply the initial (or boundary) condition, that is,

substituting

0

for

x

and

2

for

y

into the general

solution in this case, we get

Thus, the

particular

solution we are looking for is

b

.

Find the

general

solution to

Exercise:

(Waner, Problem #4, Section 7.6)

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