Chapter 8: Mathematical Modeling with
Differential Equations
Summary:
This chapter brings together the two important ideas of differentia
tion and integration of functions. These ideas are related by looking at equations
that involve both a function,
y
(
x
)
and some of its derivatives, (i.e.,
y
′
(
x
)
,
y
′′
(
x
)
,
etc.)
which may be solved in some cases using integration.
Topics discussed
in this chapter include how to solve first order differential equations, how to use
differential equations to model various physical situations, and how to find the
unique solution of a differential equation that is paired with an initial condition.
OBJECTIVES: After reading and working through this chapter
you should be able to do the following:
1. Determine the order of a differential equation (§8
.
1).
2. Determine if a given function is a solution of a differential equation (§8
.
1).
3. Solve a first order separable differential equation (§8
.
2).
4. Draw a slope field or direction field for a differential equation (§8
.
3).
5. Use Euler’s method to approximate the solution of a differential equation
(§8
.
3).
6. Solve a first order linear differential equation (§8
.
4).
7. Model various applications using first order differential equations (§8
.
1,
§8
.
2, and §8
.
4).
8.1
Modeling With Differential Equations
PURPOSE: To define what a differential equation is and to solve
some first order differential equations.
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This section introduces what differential equations are and what it means for a
function to solve a differential equation. Two types of differential equations are
discussed: first order linear equations and separable equations.
CAUTION: Differential equations will involve some unknown function
y
=
y
(
t
)
.
Usually this function is just written as
y
although it is still a function of some
independent variable.
A firstorder differential equation is an equation that involves only
y
, its derivative,
y
′
and an independent variable
t
. Sometimes, the independent variable will not
occur explicitly in the equation. For example,
y
′
=
y
2
, really could be written as
y
′
(
t
) =
y
(
t
)
2
.
A function,
y
=
φ
(
t
)
, is a
solution
of the differential equation if an equality results
solution
general solution
(i.e., both sides of the equation are equal) when the function is inserted into the
differential equation.
A
general solution
of a differential equation involves a
constant of integration that allows for an infinite number of different functions to
solve the differential equation. An
InitialValue Problem (IVP)
is a differential
initialvalue problem (IVP)
equation that is paired with some initial condition. A solution of an IVP must solve
the differential equation and also satisfy the initial condition(s).
Initial conditions
initial conditions
are substituted into the general solution in order to determine a unique value for
the constant of integration.
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 Spring '10
 lee
 Differential Equations

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