Master Second-Order Differential Equations: Enhance Your Engineering Skills

Question 1

Which of the following is the standard form of a second-order differential equation?

Accepted Answer

y'' - p(x)y = q(x)

Answer

y' + p(x)y = q(x)

Answer

y''' + p(x)y'' = q(x)

Answer

y'''' - p(x)y''' = q(x)

Question 2

Calculate the Wronskian of the functions: y₁ = x and y₂ = x^2

Accepted Answer

2x

Answer

x^2

Answer

x - x^2

Answer

2x^2

Question 3

Find the characteristic equation of the differential equation: y'' + 4y' + 5y = 0

Accepted Answer

r^2 + 4r + 5 = 0

Answer

r^2 - 4r + 5 = 0

Answer

(r + 2)^2 + 1 = 0

Answer

r^2 + 5 = 0

Question 4

Which method is best suited for solving nonhomogeneous second-order differential equations when the forcing function is a trigonometric function?

Accepted Answer

Method of undetermined coefficients

Answer

Variation of parameters

Answer

Laplace transforms

Answer

Frobenius method

Question 5

What is the general solution to the homogeneous differential equation y'' - 4y = 0?

Accepted Answer

y = c1e^(2x) + c2e^(-2x)

Answer

y = c1cos(2x) + c2sin(2x)

Answer

y = c1x + c2

Answer

y = c1x^2 + c2x

Question 6

What is the order of the differential equation d^4y/dx^4 - 3d^2y/dx^2 + 2y = 0?

Accepted Answer

4

Answer

2

Answer

3

Answer

6

Question 7

To solve the differential equation y'' - 4y' + 4y = e^(-2x), which method should you employ?

Accepted Answer

Variation of parameters

Answer

Undetermined coefficients

Answer

Laplace transforms

Question 8

Find the solution to the differential equation y'' + 9y = 0:

Accepted Answer

y(x) = c1 * sin(3x) + c2 * cos(3x)

Answer

y(x) = c1 * e^(3x) + c2 * e^(-3x)

Question 9

Which of the following is a requirement for the existence of a particular solution to the nonhomogeneous equation y'' + p(x)y' + q(x)y = f(x) using the method of undetermined coefficients?

Accepted Answer

f(x) must be continuous and piecewise differentiable.

Answer

f(x) must be a periodic function.

Question 10

Classify the differential equation y'' - 2y' + 5y = 0:

Accepted Answer

Homogeneous linear differential equation with constant coefficients

Answer

Nonhomogeneous linear differential equation with constant coefficients

Question 11

Solve the initial value problem y'' + 4y = 0, y(0) = 1, y'(0) = 2:

Accepted Answer

y(x) = cos(2x) + 2sin(2x)

Answer

y(x) = e^(2x) + 2e^(-2x)

Question 12

Define the Wronskian of two functions y1(x) and y2(x):

Accepted Answer

W(y1, y2) = y1*y2' - y2*y1'

Answer

W(y1, y2) = y1*y2 + y2*y1'

Question 13

Determine the order of the differential equation y'''' + 3y''' - 2y'' = 0:

Accepted Answer

4

Answer

2

Question 14

Which method can be used to solve systems of differential equations?

Accepted Answer

Gaussian elimination

Answer

Method of undetermined coefficients

Question 15

Which of the following represents the correct form of the particular solution for the non-homogeneous differential equation y'' + 4y = sin(2x) using the method of undetermined coefficients?

Accepted Answer

Yp(x) = A sin(2x) + B cos(2x)

Answer

Yp(x) = Ax + B

Answer

Yp(x) = (Ax + B) sin(2x)

Answer

Yp(x) = A sin(2x)

Question 16

The Wronskian of two functions y₁ and y₂ is defined as:

Accepted Answer

y₁y₂ - y₂y₁

Answer

y₁y₂ + y₂y₁

Answer

y₁'/y₂ - y₂'/y₁

Answer

y₁'/y₂ + y₂'/y₁

Question 17

A singular point of the differential equation y'' + p(x)y = 0 occurs at x₀ if:

Accepted Answer

p(x₀) = 0 and p'(x₀) ≠ 0

Answer

p(x₀) = 0 and p'(x₀) = 0

Answer

p(x₀) = ∞ and p'(x₀) = ∞

Answer

p(x₀) ≠ 0 and p'(x₀) = 0

Question 18

Which of the following differential equations is a Cauchy-Euler equation?

Accepted Answer

a₀x²y'' + a₁xy' + a₂y = 0

Answer

a₀y'' + a₁xy' + a₂y² = 0

Answer

a₀x²y' + a₁xy'' + a₂y = 0

Answer

a₀y'' + a₁x²y' + a₂y = 0

Question 19

Consider the equation y' + y = 0. Which of the following is the general solution?

Accepted Answer

y = c * e^(-x)

Answer

y = c * x

Answer

y = c * sin(x)

Answer

y = c * e^x

Question 20

Enhanced: Consider the differential equation y'' + 2py' + qy = 0. What is the characteristic equation?

Accepted Answer

Refined: r^2 + 2pr + q = 0

Answer

r^2 + 4pr + q = 0

Answer

r^2 - 2pr + q = 0

Answer

r^2 - 4pr + q = 0

Question 21

Simplified: Consider the differential equation y'' - 2y' + y = 0. What is the Laplace transform of the solution?

Accepted Answer

Simplified: Y(s) = (s - 1)/(s^2 - 2s + 1)

Answer

Y(s) = (s + 1)/(s^2 - 2s + 1)

Answer

Y(s) = (s - 1)/(s^2 + 2s + 1)

Answer

Y(s) = (s + 1)/(s^2 + 2s + 1)

Question 22

Rephrased: Which of the following represents a Dirichlet boundary condition?

Accepted Answer

Rephrased: y(0) = 1 and y(L) = 0

Answer

y'' = 0

Answer

y' = 2

Answer

y'' + y' = 1

Question 23

Refined: Find the particular solution of the differential equation y'' + 2y' + 2y = cos(2x) using the method of undetermined coefficients.

Accepted Answer

Refined: y = (1 / 2)cos(2x)

Answer

y = -cos(2x)

Answer

y = (1 / 4)cos(2x)

Answer

y = sin(2x)

Question 24

Clarified: Which of the following is an example of a Cauchy problem for a second-order differential equation?

Accepted Answer

Clarified: y'' + 2y' + y = 0, y(0) = 1, y'(0) = 0

Answer

y'' + 2y' + y = 0

Answer

y'' + y = 0, y'(0) = 0

Answer

y' + 2y = 0, y(0) = 1

Question 25

Rephrased: Consider the differential equation y'' + p(x)y' + q(x)y = 0. What is the Wronskian of two solutions y₁ and y₂?

Accepted Answer

Rephrased: W(y₁, y₂) = y₁y₂' - y₂y₁'

Answer

W(y₁, y₂) = y₁y₂ + y₂y₁

Answer

W(y₁, y₂) = y₁y₁' + y₂y₂'

Answer

W(y₁, y₂) = y₁'y₂' + y₂'y₁

Question 26

Enhanced: Which of the following is a sufficient condition for the existence of a unique solution to a second-order boundary value problem?

Accepted Answer

Refined: The Wronskian of the two solutions is nonzero at all points in the interval.

Answer

The boundary conditions are specified.

Answer

The differential equation is linear.

Question 27

Clarified: What is the order of the following differential equation: y''' + 2y'' - 3y' + y = 0?

Accepted Answer

Maintained: 3

Answer

4

Answer

5

Answer

2

Question 28

Which of the following is NOT a method for solving second-order differential equations with constant coefficients?

Accepted Answer

Integration by parts

Answer

Undetermined coefficients

Answer

Laplace transforms

Answer

Variation of parameters

Question 29

For the differential equation y'' + 4y = sin(2x), the correct form of the particular solution using the method of undetermined coefficients is:

Accepted Answer

y_p = A sin(2x) + B cos(2x)

Answer

y_p = Ax^2 + Bx + C

Answer

y_p = Ae^(2x) + Be^(-2x)

Answer

y_p = A + Be^x

Question 30

In electrical engineering, an application of second-order differential equations is:

Accepted Answer

Analyzing the transient response of an RLC circuit

Answer

Predicting the trajectory of a moving object

Answer

Forecasting the growth of a population

Answer

Modeling the speed of a moving body

Question 31

For the differential equation y'' - 2y' + y = 0, what is the characteristic equation?

Accepted Answer

λ^2 - 2λ + 1 = 0

Answer

λ^2 + 2λ + 1 = 0

Answer

λ^2 - 2λ - 1 = 0

Answer

λ^2 + 2λ - 1 = 0

Question 32

The damping ratio of a second-order differential equation is:

Accepted Answer

The ratio of the damping coefficient to the critical damping coefficient

Answer

The ratio of the natural frequency to the damping coefficient

Answer

The ratio of the amplitude to the period

Question 33

For a second-order differential equation to be critically damped, what condition must be met?

Accepted Answer

The damping coefficient is equal to twice the square root of the mass times the stiffness

Answer

The damping coefficient is greater than twice the square root of the mass times the stiffness

Answer

The damping coefficient is less than twice the square root of the mass times the stiffness

Question 34

Which property describes the complementary solution of a second-order differential equation?

Accepted Answer

It is the solution to the homogeneous equation.

Answer

It is always real and positive.

Answer

It always contains trigonometric functions.

Answer

It is the solution to the non-homogeneous equation.

Question 35

Which of the following is the general form of a second-order, linear, homogeneous differential equation with constant coefficients?

Accepted Answer

y'' + ay' + by = 0

Answer

y'''' + ay''' + by'' + cy' + dy = 0

Answer

y''' + ay'' + by' + cy = 0

Answer

y^(4) + ay''' + by'' + cy' + dy = 0

Question 36

What is the order of the following differential equation: (d^3y/dx^3) + (d^2y/dx^2) - (dy/dx) + y = 0?

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 37

The Laplace transform of the function f(t) = t is:

Accepted Answer

1/s^2

Answer

s/1

Answer

s^2/1

Answer

-1/s^2

Question 38

The differential equation y'' + 2y' + y = tan(x) can be classified as:

Accepted Answer

Non-homogeneous, second-order, linear

Answer

Non-homogeneous, first-order, non-linear

Answer

Homogeneous, second-order, linear

Answer

Homogeneous, first-order, linear

Question 39

Which of the following is a particular solution to the differential equation y'' - 2y' + y = sin(x)?

Accepted Answer

(1/2)sin(x)

Answer

sin(2x)

Answer

cos(x)

Answer

(1/2)cos(x)

Question 40

What is the characteristic equation associated with the differential equation  y'' + 4y' + 3y = 0?

Accepted Answer

r² + 4r + 3 = 0

Answer

r² - 4r - 3 = 0

Answer

r² - 4r + 3 = 0

Answer

r² + 4r - 3 = 0

Question 41

A spring-mass system has a mass of 1 kg and a spring constant of 4 N/m.  The system is subjected to an external force F(t) = cos(2t). What is the natural frequency of this system?

Accepted Answer

2 rad/s

Answer

4 rad/s

Answer

1 rad/s

Answer

0.5 rad/s

Question 42

Which of these methods is NOT suitable for solving a non-homogeneous second-order differential equation?

Accepted Answer

Separation of Variables

Answer

Method of Undetermined Coefficients

Answer

Variation of Parameters

Answer

Laplace Transform Method

Question 43

The general solution of the differential equation y'' - 4y' + 4y = 0 can be expressed in the following form:

Accepted Answer

y = (c₁ + c₂t)e^(2t)

Answer

y = c₁e^(2t) + c₂e^(-2t)

Answer

y = c₁e^(t) + c₂e^(3t)

Answer

y = c₁cos(2t) + c₂sin(2t)

Question 44

What is the Laplace transform of the function f(t) = t²?

Accepted Answer

2/s³

Answer

2/s²

Answer

1/s²

Answer

1/s³

Question 45

Calculate the Wronskian for the functions y₁ = e^t and y₂ = e^(-t).

Accepted Answer

-2

Answer

0

Answer

e^(2t)

Answer

2

Question 46

A series circuit consists of an inductor with inductance L = 1H and a capacitor with capacitance C = 0.25F. What is the angular frequency of the oscillations in this circuit?

Accepted Answer

2 rad/s

Answer

1 rad/s

Answer

0.5 rad/s

Answer

4 rad/s

Question 47

In a critically damped system, the damping ratio (ζ) is:

Accepted Answer

ζ = 1

Answer

ζ < 1

Answer

ζ > 1

Answer

ζ = 0

Question 48

What is the form of the particular solution for the differential equation y'' + y = sin(t)?

Accepted Answer

yₚ = Atcos(t) + Btsin(t)

Answer

yₚ = Ate^(t)

Answer

yₚ = Acos(t) + Bsin(t)

Answer

yₚ = At²e^(t)

Question 49

Determine the inverse Laplace transform of F(s) = 1/(s² + 4s + 13).

Accepted Answer

(1/3)e^(-2t)sin(3t)

Answer

(1/3)e^(2t)sin(3t)

Answer

e^(2t)cos(3t)

Answer

e^(-2t)cos(3t)

Question 50

A series RLC circuit is described by the differential equation d²q/dt² + 2dq/dt + q = 0, where q is the charge on the capacitor. What type of damping does this circuit exhibit?

Accepted Answer

Overdamped

Answer

Critically damped

Answer

Underdamped

Answer

Undamped

Question 51

The general solution to the differential equation d²y/dx² - 4dy/dx + 4y = 0 is:

Accepted Answer

y = (A + Bx)e^(2x)

Answer

y = Ae^(x) + Be^(2x)

Answer

y = Ae^(2x) + Be^(-2x)

Answer

y = Ae^(2x)

Question 52

A mass-spring system is described by the equation d²x/dt² + 4x = 0. What is the natural frequency of the system?

Accepted Answer

2 rad/s

Answer

4 rad/s

Answer

1 rad/s

Question 53

What is the particular solution to the non-homogeneous equation d²y/dx² + y = sin(x)?

Accepted Answer

y = (-1/2)xcos(x)

Answer

y = (1/2)xsin(x)

Answer

y = cos(x)

Answer

y = sin(x)

Question 54

Which of the following methods can be used to solve the differential equation d²y/dx² + 4dy/dx + 3y = e^(-x)?

Accepted Answer

Variation of Parameters

Answer

Laplace Transform

Answer

All of the above

Answer

Method of Undetermined Coefficients

Question 55

The Laplace transform of the function f(t) = t² is:

Accepted Answer

2/s³

Answer

2/s²

Answer

1/s³

Answer

1/s²

Question 56

Using the Laplace transform, the solution to the initial value problem d²y/dt² + y = 0, y(0) = 1, y'(0) = 0 is:

Accepted Answer

y = cos(t)

Answer

y = sin(t)

Answer

y = e^(-t)

Answer

y = e^(t)

Question 57

A second-order differential equation is considered to be linear if:

Accepted Answer

The dependent variable and its derivatives appear only in a linear combination.

Answer

The coefficients are constant.

Answer

The independent variable appears only in a linear combination.

Question 58

The characteristic equation of a second-order differential equation with constant coefficients determines:

Accepted Answer

The form of the general solution

Answer

The initial conditions of the solution

Answer

The specific values of the constants in the general solution

Answer

The particular solution

Question 59

A system is considered to be stable if:

Accepted Answer

Its response to any bounded input remains bounded.

Answer

Its response to any input returns to its initial state.

Answer

Its output remains constant over time.

Answer

Its input and output are directly proportional.