1016 Chapter 16: Additional Topics in Differential Equations
____ 6. Solve the differential equation by the method of undetermined
coefficients.
a.
b.
c.
d.
e.
____ 7. Solve the differential equation by the method of undetermined coefficients.
a.
b.
c.
d.
e.
16.3 Second-Order Nonhomogeneous Linear Equations
1017
____
8.
Solve the differential equation
coefficients.
a.
b.
c.
d.
e.
____
9.
Solve the differential equation
, where
method of undetermined coefficients.
a.
b.
c.
d.
e.
____
10.
Solve the differential equation
by the method of undetermined coefficients.
a.
b.
c.
d.
e.
1018 Chapter 16: Additional Topics in Differential Equations
____
11.
Solve the differential equation
by the method of variation of
parameters.
a.
b.
c.
d.
e.
____
12.
Solve the differential equation
by the method of variation of
parameters.
a.
b.
c.
d.
e.
____
13.
Using the method of undetermined coefficients, determine the most suitable choice
for
given
. (You do not need to solve the differential equation.)
a.
b.
c.
d.
e.
16.3 Second-Order Nonhomogeneous Linear Equations
1019
____ 14.
Use the electrical circuit differential equation
where
is the resistance (in ohms),
is the capacitance (in farads),
is the
inductance (in henrys), is the electromotive force (in volts), and q is the charge on the
capacitor (in coulombs). Find the charge q as a function of time for the electrical circuit described.
Assume that and .
a.
b.
c.
d.
e.
____ 15. Find the particular solution of the differential equation
for the oscillating motion of an object on the end of a spring. In
the equation, y is the displacement from equilibrium (positive direction is downward) measured in
feet, and t is the time in seconds (see figure). The constant is the weight of the object,
is the acceleration due to gravity, is the magnitude of the resistance to the motion, is the
spring constant from Hooke’s Law, is the acceleration imposed on the system,
and .
a.
1020 Chapter 16: Additional Topics in Differential Equations
b.
c.
d.
e.
____ 16. Find the particular solution of the differential equation
for the oscillating motion of an object on the end of a spring. In
the equation, y is the displacement from equilibrium (positive direction is downward) measured in
feet, and t is the time in seconds (see figure). The constant is the weight of the object,
is the acceleration due to gravity, is the magnitude of the resistance to the motion, is the
spring constant from Hooke’s Law, is the acceleration imposed on the system,
and .
a.
b.
c.
d.
e.
16.3 Second-Order Nonhomogeneous Linear Equations
1021
____ 17. Find the particular solution of the differential equation
for the oscillating motion of an object on the end of a spring. In
the equation, y is the displacement from equilibrium (positive direction is downward) measured in
feet, and t is the time in seconds (see figure). The constant is the weight of the object,
is the acceleration due to gravity, is the magnitude of the resistance to the motion, is
the spring constant from Hooke’s Law, is the acceleration imposed on the system,
and .
a.
b.
c.
d.
e.
1022
Chapter 16: Additional Topics in Differential Equations
____
18.
Solve the differential equation
given that
and
are solutions of the corresponding homogeneous equation.
a.
b.
c.
d.
e.
16.3 Second-Order Nonhomogeneous Linear Equations
1023
16.3 Second-Order Nonhomogeneous Linear Equations
Answer Section
1024 Chapter 16: Additional Topics in Differential Equations
16.4 Series Solutions of Differential Equations
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
a.
b.
c.
d.
e.
____ 2.
a.
b.
c.
d.
e.
____ 3.
a.
b.
c.
d.
e.
____ 4.
a.
b.
Use a power series to solve the differential equation .
where is an arbitrary constant
where is an arbitrary constant
where is an arbitrary constant
where is an arbitrary constant ,
where is an arbitrary constant
Use a power series to solve the differential equation .
where and are arbitrary constants
where and are arbitrary constants
where and are arbitrary constants
where and are arbitrary constants
where and are arbitrary constants
Use a power series to solve the differential equation .
where and are arbitrary constants
whereand are arbitrary constants
, where and are arbitrary constants
, where and are arbitrary constants
, where and are arbitrary constants
Solve the differential equation .
where is an arbitrary constant
where is an arbitrary constant
16.4 Series Solutions of Differential Equations
1025
c.
, where is an arbitrary constant
d.
, where
is an arbitrary constant
e.
, where
is an arbitrary constant
____
5.
Find the interval of convergence for the solution of the differential equation
.
a.
b.
c.
d.
e.
____
6.
Solve the differential equation
.
a.
, where and are arbitrary constants
b.
, where and are arbitrary constants
c.
, where and are arbitrary constants
d.
, where and are arbitrary constants
e.
, where and are arbitrary constants
1026 Chapter 16: Additional Topics in Differential Equations
____ 7. Find the interval of convergence for the solution of the differential equation
.
a.
b.
c.
d.
e.
____ 8. Solve the differential equation .
a.
, where is an arbitrary constant
b.
, where and are arbitrary constants
c.
, where and are arbitrary
constants
d.
, where and are arbitrary
constants
e.
, where and are arbitrary constants
____ 9. Find the interval of convergence for the solution of the differential equation
.
a.
b.
c.
d.
e.
16.4 Series Solutions of Differential Equations
1027
____
10.
Find the first six terms of the power series representing independent solutions of the
differential equation
.
a.
, where
and
are arbitrary
constants
b.
, where
and
are arbitrary
constants
c.
, where
and
are arbitrary
constants
d.
, where
and
are arbitrary
constants
e.
, where
and
are arbitrary
constants
____
11.
Find the first eight terms of the power series representing independent solutions of
the differential equation
.
a.
, where
and
are
arbitrary constants
1028 Chapter 16: Additional Topics in Differential Equations
b.
where and are arbitrary constants
c.
,
where and are arbitrary constants
d.
, where and are
arbitrary constants
e.
,
where and are arbitrary constants
____ 12. Use Taylor’s Theorem to find the first five terms of the series solution of
given the initial condition .
a.
b.
c.
d.
e.
16.4 Series Solutions of Differential Equations
1029
____
13.
Use Taylor’s Theorem to find the first four terms of the series solution of
given the initial condition
and use it to calculate
. Round your answer to three decimal
places wherever applicable.
a.
b.
c.
d.
e.
____
14.
Find the series solution of the differential equation
given the initial
conditions
and
.
a.
b.
c.
d.
e.
1030
Chapter 16: Additional Topics in Differential Equations
____
15.
Use Taylor’s Theorem to find the first six terms of the series solution of
given the initial conditions
and
.
a.
b.
c.
d.
e.
____ 16. Use Taylor’s Theorem to find the first eight terms of the series solution of
given the initial conditions and use it to calculate .
Round your answer to three decimal places.
a.
b.
c.
d.
e.
16.4 Series Solutions of Differential Equations
1031
____ 17.
Use Taylor’s Theorem to find the first four terms of the series solution of
given the initial conditions
.
a.
b.
c.
d.
e.
____ 18. Use Taylor’s Theorem to find the first four terms of the series solution of
given the initial conditions and use it to calculate
. Round your answer to three decimal places.
a.
b.
c.
d.
e.
1032 Chapter 16: Additional Topics in Differential Equations
16.4 Series Solutions of Differential Equations
Answer Section