Question 1

Identify the form of a particular solution to the equation   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)    A

Question 2

Solve the initial value problem   ,   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)    D

Question 3

Solve the initial value problem   ,   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)    B

Question 4

Find the general solution of   , given the particular solution   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 5

Identify the pair of graphs that correspond most closely to the solutions of   with   , respectively. [The function y(t) is plotted on the vertical axes and t is plotted on the horizontal axes.]

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 6

For a pendulum of weight 4 pounds, length 0.60 ft, damping constant   and forcing function   find the amplitude and period of the steady-state motion. [The acceleration due to gravity is 32 ft s-2.]

Accepted Answer

A)  amplitude: 0.400 ft; period: 1.555 rad 
B)  amplitude: 0.200 ft; period: 0.777 rad 
C)  amplitude: 0.092 ft; period: 0.524 rad 
D)  amplitude: 0.184 ft; period: 1.047 rad 
A)  amplitude: 0.400 ft; period: 1.555 rad 
B)  amplitude: 0.200 ft; period: 0.777 rad 
C)  amplitude: 0.092 ft; period: 0.524 rad 
D)  amplitude: 0.184 ft; period: 1.047 rad

Question 7

Solve the initial value problem   ,   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 8

Solve the initial value problem   ,   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 9

A 2.0 kg mass hangs on a spring with a 0.5 newton/meter force constant and its motion is not damped. If the system is subjected to an external variable-frequency vibration described as   newtons, at what frequency,   , will the external vibration and the spring system be in resonance?

Accepted Answer

A)  0.25 s-1 
B)  0.50 s-1 
C)  4.00 s-1 
D)  2.00 s-1 
A)  0.25 s-1 
B)  0.50 s-1 
C)  4.00 s-1 
D)  2.00 s-1

Question 10

For   find the steady-state solution and identify its amplitude and phase shift.

Accepted Answer

@#IMG-DLM& ; amplitu

Question 11

A series circuit has a 0.25 henry inductor, a 500 ohm resistor, and a 0.000004 farad capacitor. There is an initial charge of 0.000001 coulombs, there is no initial current, and there is an applied voltage which is described as   . Identify the solution to the differential equation that describes the charge on the capacitor as a function of time.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 12

A series circuit has a 0.1 henry inductor, a 310 ohm resistor, and a 0.000008 farad capacitor. There is an initial charge of 0.000003 coulombs, there is no initial current, and there is an applied voltage which is described as   . Identify the solution to the differential equation that describes the charge on the capacitor as a function of time.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 13

Find the general solution of the differential equation.

Accepted Answer

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B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 14

Identify the general solution of the the differential equation   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 15

Identify the radius of convergence of the power series solutions about x = 0 of   .

Accepted Answer

A)  R = 0 
B)    
C)  R =   
D)  R = 8 
A)  R = 0 
B)    
C)  R =   
D)  R = 8

Question 16

A pendulum has length 0.22 meter. A bob is released from rest from a starting angle   . Find an equation for the position at any time t and find the amplitude and period of the motion. [The acceleration due to gravity is 9.8 m s-2.]

Accepted Answer

A)  amplitude: 0.2 m; period: 0.941 rad 
B)  amplitude: 0.2 m; period: 0.471 rad 
C)  amplitude: 0.4 m; period: 0.471 rad 
D)  amplitude: 0.4 m; period: 0.941 rad 
A)  amplitude: 0.2 m; period: 0.941 rad 
B)  amplitude: 0.2 m; period: 0.471 rad 
C)  amplitude: 0.4 m; period: 0.471 rad 
D)  amplitude: 0.4 m; period: 0.941 rad

Question 17

A second order differential equaiton can be arranged to the form   , and one can find the third and higher derivatives of y by simply differentiating this equation. Since a Taylor series expansion of a function y(x) is   , one can differentiate the rearranged second order differential equation to evaluate coefficients of the Taylor polynomial, if one is either given or can solve for the initial condition y(0) and y'(0). What does the fourth-degree Taylor polynomial look like for the solution to the equation   if the initial conditions are   ?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 18

Consider solutions to the second order differential equation   in which m, c, and k are positive constants. Which of the following pairs of graphs might correspond to the sets of constants m = 1.00, c = 0.80, and k = 1.20 (blue graph), and m = 1.00, c = 0.08, and k = 1.20 (red graph) ?

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 19

A series circuit has a 0.1 henry inductor, a 490 ohm resistor, and a 0.000003 farad capacitor. There is an initial charge of 0.000009 coulombs, there is no initial current, and there is an applied voltage which is described as   . Identify the steady-state solution to the differential equation.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 20

Find the general solution of   , given the particular solution   .

Accepted Answer

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B)    
C)    
D)    
A)    
B)    
C)    
D)

Identify the form of a particular solution to the equation .

Solve the initial value problem , .

Solve the initial value problem , .

Find the general solution of , given the particular solution .

Identify the pair of graphs that correspond most closely to the solutions of with , respectively. [The function y(t) is plotted on the vertical axes and t is plotted on the horizontal axes.]

For a pendulum of weight 4 pounds, length 0.60 ft, damping constant and forcing function find the amplitude and period of the steady-state motion. [The acceleration due to gravity is 32 ft s^-2.]

Solve the initial value problem , .

Solve the initial value problem , .

A 2.0 kg mass hangs on a spring with a 0.5 newton/meter force constant and its motion is not damped. If the system is subjected to an external variable-frequency vibration described as newtons, at what frequency, , will the external vibration and the spring system be in resonance?

For find the steady-state solution and identify its amplitude and phase shift.

Find the general solution of the differential equation.

Identify the general solution of the the differential equation .

Identify the radius of convergence of the power series solutions about x = 0 of .

A pendulum has length 0.22 meter. A bob is released from rest from a starting angle . Find an equation for the position at any time t and find the amplitude and period of the motion. [The acceleration due to gravity is 9.8 m s^-2.]

Consider solutions to the second order differential equation in which m, c, and k are positive constants. Which of the following pairs of graphs might correspond to the sets of constants m = 1.00, c = 0.80, and k = 1.20 (blue graph), and m = 1.00, c = 0.08, and k = 1.20 (red graph) ?

A series circuit has a 0.1 henry inductor, a 490 ohm resistor, and a 0.000003 farad capacitor. There is an initial charge of 0.000009 coulombs, there is no initial current, and there is an applied voltage which is described as . Identify the steady-state solution to the differential equation.

Find the general solution of , given the particular solution .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Filters

Exam 17: Second Order Differential Equations

Identify the form of a particular solution to the equation .

Solve the initial value problem , .

Solve the initial value problem , .

Find the general solution of , given the particular solution .

Identify the pair of graphs that correspond most closely to the solutions of with , respectively. [The function y(t) is plotted on the vertical axes and t is plotted on the horizontal axes.]

For a pendulum of weight 4 pounds, length 0.60 ft, damping constant and forcing function find the amplitude and period of the steady-state motion. [The acceleration due to gravity is 32 ft s-2.]

Solve the initial value problem , .

Solve the initial value problem , .

A 2.0 kg mass hangs on a spring with a 0.5 newton/meter force constant and its motion is not damped. If the system is subjected to an external variable-frequency vibration described as newtons, at what frequency, , will the external vibration and the spring system be in resonance?

For find the steady-state solution and identify its amplitude and phase shift.

Find the general solution of the differential equation.

Identify the general solution of the the differential equation .

Identify the radius of convergence of the power series solutions about x = 0 of .

A pendulum has length 0.22 meter. A bob is released from rest from a starting angle . Find an equation for the position at any time t and find the amplitude and period of the motion. [The acceleration due to gravity is 9.8 m s-2.]

Consider solutions to the second order differential equation in which m, c, and k are positive constants. Which of the following pairs of graphs might correspond to the sets of constants m = 1.00, c = 0.80, and k = 1.20 (blue graph), and m = 1.00, c = 0.08, and k = 1.20 (red graph) ?

A series circuit has a 0.1 henry inductor, a 490 ohm resistor, and a 0.000003 farad capacitor. There is an initial charge of 0.000009 coulombs, there is no initial current, and there is an applied voltage which is described as . Identify the steady-state solution to the differential equation.

Find the general solution of , given the particular solution .

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Filters

For a pendulum of weight 4 pounds, length 0.60 ft, damping constant and forcing function find the amplitude and period of the steady-state motion. [The acceleration due to gravity is 32 ft s^-2.]

A pendulum has length 0.22 meter. A bob is released from rest from a starting angle . Find an equation for the position at any time t and find the amplitude and period of the motion. [The acceleration due to gravity is 9.8 m s^-2.]