Deck 17: Second Order Differential Equations

Question 1

A 0.15 kg mass hangs on a spring with a 2 N m^-1 force constant and its motion is damped proportional to its velocity with proportionality constant 0.4 kg s^-1. If the system is subjected to an external variable-frequency vibration described as newtons, what will be the amplitude of the steady-state oscillation?

A) 1.32 m
B) 1.74 m
C) 0.53 m
D) 0.00 m

Answer 1

1.32 m

Question 2

Solve the initial value problem.

Answer 2

Question 3

Solve the initial value problem , .

A)
B)
C)
D)

Answer 3

Question 4

Find the general solution of the differential equation.

A)
B)
C)
D)

Question 5

Find the general solution of the equation .

A)
B)
C)
D)

Question 6

Identify the general solution of the the differential equation .

A)
B)
C)
D)

Question 7

Solve the initial value problem , .

A)
B)
C)
D)

Question 8

Identify the general solution of the the differential equation .

A)
B)
C)
D)

Question 9

Identify the form of a particular solution to the equation .

A)
B)
C)
D)

Question 10

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?

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

Question 11

Find the general solution of , given the particular solution .

A)
B)
C)
D)

Question 12

Determine the form of a particular solution of the equation.

A)
B)
C)
D)

Question 13

A certain spring is at rest when stretched 0.098 m by a 2.0 kg mass. Which function describes the motion of the mass if it is pulled down 0.25 m and released without imparting any initial velocity at time t = 0? Other helpful information: the motion is not damped; use 9.8 m s^-2 as the acceleration due to gravity; and consider the zero position to be the rest position of the spring with the mass attached, and downward motion defines the positive x direction.

A)
B)
C)
D)

Question 14

Identify the general solution of the the differential equation .

A)
B)
C)
D)

Question 15

Find the general solution of , given the particular solution .

A)
B)
C)
D)

Question 16

Solve the initial value problem , .

A)
B)
C)
D)

Question 17

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)
B)
C)
D)

Question 18

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

Question 19

A spring is stretched 2 cm by a 1-kg mass. The mass is set in motion from its equilibrium position with an upward velocity of 4 m/s. The damping constant equals Find an equation for the position of the mass at any time t.

Question 20

Identify the form of a particular solution to the equation .

A)
B)
C)
D)

Question 21

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.

A)
B)
C)
D)

Question 22

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.]

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 23

Suppose that the charge in a circuit satisfies the equation Find the gain of the circuit.

A)
B)
C)
D)

Question 24

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.

A)
B)
C)
D)

Question 25

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 ?

A)
B)
C)
D)

Question 26

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.

A)
B)
C)
D)

Question 27

Find the recurrence relation for the general power series solution to the second order equation .

A)
B)
C)
D)

Question 28

Solve the initial value problem , .

A)
B)
C)
D)

Question 29

A series circuit has an 0.3 henry inductor, a 300 ohm resistor, and a 10^-3 farad capacitor. The initial charge on the capacitor is 10^-7 coulombs, and there is no initial current nor applied voltage. Identify the function that describes the charge on the capacitor as a function of time.

A)
B)
C)
D)

Question 30

Find the recurrence relation for the general power series solution to the second order equation .

A)
B)
C)
D)

Question 31

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

A)
B)
C)
D)

Question 32

A series circuit has a 0.25 henry inductor, a 340 ohm resistor, and a 0.000009 farad capacitor. There is an initial charge of 0.000004 coulombs, there is no initial current, and there is an applied voltage which is described as . Identify the amplitude of the steady-state solution.

A) 0.000003
B) 0.000009
C) 0.35
D) 0.175

Question 33

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) 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 34

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

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

Question 35

Solve the initial value problem , .

A)
B)
C)
D)

Question 36

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.]

A)
B)
C)
D)

Question 37

A series circuit has a 0.15 henry inductor, a 490 ohm resistor, and a 0.000004 farad capacitor. There is an applied voltage which is described as . Identify the general solution to the differential equation that describes the charge on the capacitor as a function of time.

A)
B)
C)
D)

Question 38

A second order differential equation 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 in the Taylor polynomial, if one is either given or can solve for the initial condition y(0) and y'(0). What is the coefficient of x⁴ in the Taylor polynomial expansion of the solution to the equation if the initial conditions are ?

A)
B)
C)
D)