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MAGENTA
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Larson Texts, Inc. • Multivariable Calculus 11e • CALC11-WFH
16.1
Exact First-Order Equations
16.2
Second-Order Homogeneous Linear Equations
16.3
Second-Order Nonhomogeneous Linear Equations
16.4
Series Solutions of Differential Equations
Electrical Circuits
(Exercises 29 and 30, p. 1151)
Parachute Jump
(Section Project, p. 1152)
Motion of a Spring
(Example 8, p. 1142)
Cost
(Exercise 45, p. 1136)
Undamped or Damped Motion?
(Exercise 53, p. 1144)
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Final Pages
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Additional Topics in
Differential Equations
© Cengage Learning.
Not for distribution.

1130
Chapter 16
Additional Topics in Differential Equations
16.1
Exact First-Order Equations
Solve an exact differential equation.
Use an integrating factor to make a differential equation exact.
Exact Differential Equations
In Chapter 6, you studied applications of differential equations to growth and decay
problems. You also learned more about the basic ideas of differential equations and
studied the solution technique known as separation of variables. In this chapter, you will
learn more about solving differential equations and using them in real-life applications.
This section introduces you to a method for solving the first-order differential equation
M
(
x
,
y
)
dx
+
N
(
x
,
y
)
dy
=
0
for the special case in which this equation represents the exact differential of a
function
z
=
f
(
x
,
y
)
.
Definition of an Exact Differential Equation
The equation
M
(
x
,
y
)
dx
+
N
(
x
,
y
)
dy
=
0
is an
exact differential equation
when there exists a function
f
of two variables
x
and
y
having continuous partial derivatives such that
f
x
(
x
,
y
)
=
M
(
x
,
y
)
and
f
y
(
x
,
y
)
=
N
(
x
,
y
)
.
The general solution of the equation is
f
(
x
,
y
)
=
C
.
From Section 13.3, you know that if
f
has continuous second partials, then
∂
M
∂
y
=
∂
2
f
∂
y
∂
x
=
∂
2
f
∂
x
∂
y
=
∂
N
∂
x
.
This suggests the following test for exactness.
THEOREM 16.1
Test for Exactness
Let
M
and
N
have continuous partial derivatives on an open disk
R
. The
differential equation
M
(
x
,
y
)
dx
+
N
(
x
,
y
)
dy
=
0
is exact if and only if
∂
M
∂
y
=
∂
N
∂
x
.
Every differential equation of the form
M
(
x
)
dx
+
N
(
y
)
dy
=
0
is exact. In other words, a separable differential equation is actually a special type of
an exact equation.
Exactness is a fragile condition in the sense that seemingly minor alterations in
an exact equation can destroy its exactness. This is demonstrated in the next example.
9781337275378_1601
09/13/16
Final Pages
CYAN
MAGENTA
YELLOW
BLACK
Larson Texts, Inc. • Multivariable Calculus 11e • CALC11-WFH
© Cengage Learning.
Not for distribution.

16.1
Exact First-Order Equations
1131
Testing for Exactness
Determine whether each differential equation is exact.