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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