differential_equations - inaccessible slide presentation - Science ...

stickshrivelMechanics

Oct 24, 2013 (3 years and 7 months ago)

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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)