Question 1

Find .

Accepted Answer

A)  18   
B)  - 18   - 72   + 90   
C)    - 2   +   
D)  - 18   - 72   + 90   
E)    - 2   +   
A)  18   
B)  - 18   - 72   + 90   
C)    - 2   +   
D)  - 18   - 72   + 90   
E)    - 2   +   E

Question 2

Write the single differential equation   - 3   + 5x = 0 as an equivalent linear system using the change of variables x = u and   = v.

Accepted Answer

(i) x = u ...... (1)
Differentiating both sides of (1) with respect to t we obtain
   =   . Now   = v ;
hence   = v ........... (2)
Next   = v ...... (3)

Question 3

Find the eigenvalues of the linear system   .

Accepted Answer

A)  -3, -8 
B)  2, -12 
C)  -2, 12 
D)  6, 4 
E)  -6, -4 
A)  -3, -8 
B)  2, -12 
C)  -2, 12 
D)  6, 4 
E)  -6, -4 D

Question 4

Use the Runge-Kutta method with step size h = 0.1 to approximate the solution of the initial-value problem     on the interval

Accepted Answer

A)  0.5321 
B)  0.5380 
C)  0.5407 
D)  0.5487 
E)  0.5582 
A)  0.5321 
B)  0.5380 
C)  0.5407 
D)  0.5487 
E)  0.5582

Question 5

Use the improved Euler method to determine an approximate value of the solution of the initial-value problem     using step size

Accepted Answer

A)  27.3480 
B)  40.9407 
C)  59.9382 
D)  88.7036 
E)  64.8978 
A)  27.3480 
B)  40.9407 
C)  59.9382 
D)  88.7036 
E)  64.8978

Question 6

An LCR circuit consists of a resistor R (having resistance R measured in ohms), a capacitor C (having capacitance C measured in farads), and an inductor L (having inductance L measured in henries) and connected in a series together with an electromotive force (such as a battery or generator) E (measured in volts V).By Kirchhoff's law, the current i (measured in amperes) satisfies the differential equation L   + Ri +     = E(t).
Let L = 10 henries, R = 0 ohms, C = 0.1 farad, and E(t) = 20 cos(t).

Determine the current i at any time t ≥ 0 if the current is zero at t = 0.

Accepted Answer

i = t cos(

Question 7

Solve the differential equation   = 6x   .

Accepted Answer

A)  x = C   
B)  x = C   
C)  x = C   
D)  x = C   
E)  x= C   
A)  x = C   
B)  x = C   
C)  x = C   
D)  x = C   
E)  x= C

Question 8

Solve the differential equation (3y + 8x   ) dx + (x + 4   y) dy = 0 by finding an integrating factor depending on x.

Accepted Answer

A)  3xy + 4     = C 
B)    y + 3     = C 
C)    y + 2     = C 
D)    y + 4     = C 
E)    y + 2     = C 
A)  3xy + 4     = C 
B)    y + 3     = C 
C)    y + 2     = C 
D)    y + 4     = C 
E)    y + 2     = C

Question 9

State the order of the following differential equation and whether it is linear or nonlinear:   = xy.

Accepted Answer

A)  1st order; nonlinear 
B)  1st order; linear 
C)  2nd order; nonlinear 
D)  2nd order; linear 
E)  None of the above 
A)  1st order; nonlinear 
B)  1st order; linear 
C)  2nd order; nonlinear 
D)  2nd order; linear 
E)  None of the above

Question 10

State the order of the following differential equation and whether it is linear or nonlinear:   = 3x.

Accepted Answer

A)  1st order; linear 
B)  1st order; nonlinear 
C)  6th order; linear 
D)  6th order; nonlinear 
E)  None of the above 
A)  1st order; linear 
B)  1st order; nonlinear 
C)  6th order; linear 
D)  6th order; nonlinear 
E)  None of the above

Question 11

Let F(t) =   .Express F(t) in terms of the Heaviside function H(t -a

Accepted Answer

A)  F(t) = 9 + 11 H(t - 4) - H(t - 7) 
B)  F(t) = 9 + 7 H(t - 4) + 5 H(t - 7) 
C)  F(t) = 9 + 2 H(t + 4) - 3 H(t + 7) 
D)  F(t) = 9 + 2 H(t - 4) - 3 H(t - 7) 
E)  F(t) = 9 - 7 H(t - 4) - 5 H(t - 7) 
A)  F(t) = 9 + 11 H(t - 4) - H(t - 7) 
B)  F(t) = 9 + 7 H(t - 4) + 5 H(t - 7) 
C)  F(t) = 9 + 2 H(t + 4) - 3 H(t + 7) 
D)  F(t) = 9 + 2 H(t - 4) - 3 H(t - 7) 
E)  F(t) = 9 - 7 H(t - 4) - 5 H(t - 7)

Question 12

Find the general solution of the differential equation   - 10   + 74y = 0.

Accepted Answer

A)  y =     cos(7t) +     sin(7t) 
B)  y =     cos(7t) +     sin(7t) 
C)  y =     cos(5t) +     sin(5t) 
D)  y =     cos(5t) +     sin(5t) 
E)  y =     cos(7t) +     sin(7t) 
A)  y =     cos(7t) +     sin(7t) 
B)  y =     cos(7t) +     sin(7t) 
C)  y =     cos(5t) +     sin(5t) 
D)  y =     cos(5t) +     sin(5t) 
E)  y =     cos(7t) +     sin(7t)

Question 13

The change of dependent variable y =   transforms the equation x   + y = x   cos(x) into

Accepted Answer

A)    +   =   
B)  ln(v)   +   =   
C)    +   = cos(x) 
D)    -   = cos(x) 
E)    -   = -cos(x) 
A)    +   =   
B)  ln(v)   +   =   
C)    +   = cos(x) 
D)    -   = cos(x) 
E)    -   = -cos(x)

Question 14

Solve the following initial-value problem:   = 9.8 - 0.196v, v(0) = 48.

Accepted Answer

A)  v = 50 -   
B)  v = 50 - 2   
C)  v = 50 + 2   
D)  v = 100 - 4   
E)  v = 50 +   
A)  v = 50 -   
B)  v = 50 - 2   
C)  v = 50 + 2   
D)  v = 100 - 4   
E)  v = 50 +

Question 15

The fixed point at the origin of the autonomous linear system   is

Accepted Answer

A)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v = - u and the alpha limit for the trajectory v =   u. 
B)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v =   u and the alpha limit for the trajectory v = - u. 
C)  a stable node with the fixed point at the origin being the limit point of all trajectories. 
D)  an unstable node with the fixed point at the origin being the limit point of all trajectories. 
E)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v = u and the alpha limit for the trajectory v = 2 u. 
A)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v = - u and the alpha limit for the trajectory v =   u. 
B)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v =   u and the alpha limit for the trajectory v = - u. 
C)  a stable node with the fixed point at the origin being the limit point of all trajectories. 
D)  an unstable node with the fixed point at the origin being the limit point of all trajectories. 
E)  a saddle point with the fixed point (0 , 0) being the omega limit for the trajectory v = u and the alpha limit for the trajectory v = 2 u.

Question 16

A body of mass m =   (kg) is attached to the end of a spring with Hooke's constant k = 50 (N/m) and is set in motion with initial position x(0) =   (m) and initial velocity   (0) = - 10 (m/s). (The mass is displaced to the right and moving to the left at time t = 0.)Find (a) the position x(t) of the body at later times t, (b) the amplitude, (c) frequency, and (d) period of its oscillation.

Accepted Answer

(a) x(t) = @#IMG-DLM& cos(10 t

Question 17

Find a homogeneous linear second-order differential equation with constant coefficient having general solution y =     +   t   .

Accepted Answer

A)    + 20   - 100y = 0 
B)    + 20   + 100y = 0 
C)    - 20   + 100y = 0 
D)    - 20   - 100y = 0 
E)    + 10   + 100y = 0 
A)    + 20   - 100y = 0 
B)    + 20   + 100y = 0 
C)    - 20   + 100y = 0 
D)    - 20   - 100y = 0 
E)    + 10   + 100y = 0

Question 18

Find a particular solution of the differential equation   + y = 8sin x - 6 cos(x).

Accepted Answer

A)    = -3xcos(x) - 4x sin(x) 
B)    = 3xcos(x) + 4x sin(x) 
C)    = -4cos(x) - 3sin(x) 
D)    = 4xcos(x) + 3x sin(x) 
E)    = -4xcos(x) - 3x sin(x) 
A)    = -3xcos(x) - 4x sin(x) 
B)    = 3xcos(x) + 4x sin(x) 
C)    = -4cos(x) - 3sin(x) 
D)    = 4xcos(x) + 3x sin(x) 
E)    = -4xcos(x) - 3x sin(x)

Question 19

Find F(t) ,the inverse Laplace transform of f(s) =   -   +   .

Accepted Answer

A)  F(t) =   
B)  F(t) =   
C)  F(t) =   
D)  F(t) =   
E)  F(t) =   
A)  F(t) =   
B)  F(t) =   
C)  F(t) =   
D)  F(t) =   
E)  F(t) =

Question 20

Find the solution of the initial value problem   + 4y = F(t), y(0) = 0 ,   (0) = - 1, whereF(t) =   .

Accepted Answer

Let Y(s) =

Find .

Write the single differential equation - 3 + 5x = 0 as an equivalent linear system using the change of variables x = u and = v.

Find the eigenvalues of the linear system .

Use the Runge-Kutta method with step size h = 0.1 to approximate the solution of the initial-value problem on the interval

Use the improved Euler method to determine an approximate value of the solution of the initial-value problem using step size

Solve the differential equation = 6x .

Solve the differential equation (3y + 8x ) dx + (x + 4 y) dy = 0 by finding an integrating factor depending on x.

State the order of the following differential equation and whether it is linear or nonlinear: = xy.

State the order of the following differential equation and whether it is linear or nonlinear: = 3x.

Let F(t) = .Express F(t) in terms of the Heaviside function H(t -a

Find the general solution of the differential equation - 10 + 74y = 0.

The change of dependent variable y = transforms the equation x + y = x cos(x) into

Solve the following initial-value problem: = 9.8 - 0.196v, v(0) = 48.

The fixed point at the origin of the autonomous linear system is

Find a homogeneous linear second-order differential equation with constant coefficient having general solution y = + t .

Find a particular solution of the differential equation + y = 8sin x - 6 cos(x).

Find F(t) ,the inverse Laplace transform of f(s) = - + .

Find the solution of the initial value problem + 4y = F(t), y(0) = 0 , (0) = - 1, whereF(t) = .

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Differentiation

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Exam 19: Ordinary Differential Equations

Find .

Write the single differential equation - 3 + 5x = 0 as an equivalent linear system using the change of variables x = u and = v.

Find the eigenvalues of the linear system .

Use the Runge-Kutta method with step size h = 0.1 to approximate the solution of the initial-value problem on the interval

Use the improved Euler method to determine an approximate value of the solution of the initial-value problem using step size

Solve the differential equation = 6x .

Solve the differential equation (3y + 8x ) dx + (x + 4 y) dy = 0 by finding an integrating factor depending on x.

State the order of the following differential equation and whether it is linear or nonlinear: = xy.

State the order of the following differential equation and whether it is linear or nonlinear: = 3x.

Let F(t) = .Express F(t) in terms of the Heaviside function H(t -a

Find the general solution of the differential equation - 10 + 74y = 0.

The change of dependent variable y = transforms the equation x + y = x cos(x) into

Solve the following initial-value problem: = 9.8 - 0.196v, v(0) = 48.

The fixed point at the origin of the autonomous linear system is

Find a homogeneous linear second-order differential equation with constant coefficient having general solution y = + t .

Find a particular solution of the differential equation + y = 8sin x - 6 cos(x).

Find F(t) ,the inverse Laplace transform of f(s) = - + .

Find the solution of the initial value problem + 4y = F(t), y(0) = 0 , (0) = - 1, whereF(t) = .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Filters