Higher-Order Differential Equations: Interactive Learning Quizzes

Question 1

Determine if the functions y₁ = e^x and y₂ = e^(2x) form a fundamental set of solutions to the differential equation y'' - 3y' + 2y = 0.

Accepted Answer

Yes, because they are linearly independent.

Answer

No, because they are not linearly independent.

Answer

Yes, because they satisfy the initial conditions y(0) = 1 and y'(0) = 0.

Answer

No, because they do not satisfy the initial conditions y(0) = 1 and y'(0) = 0.

Question 2

Find the characteristic equation of the differential equation y''' + 3y'' + 3y' + y = 0.

Accepted Answer

r³ + 3r² + 3r + 1 = 0

Answer

r³ - 3r² + 3r - 1 = 0

Answer

(r + 1)³ = 0

Answer

(r - 1)³ = 0

Question 3

Which of the following differential equations can be solved using the method of variation of parameters?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

If the Wronskian of two functions y₁ and y₂ is W(y₁, y₂) = 0 at some point x, then what can be concluded?

Accepted Answer

y₁ and y₂ are linearly dependent.

Answer

y₁ and y₂ are linearly independent.

Answer

y₁ = 0 or y₂ = 0 at x.

Answer

y₁ and y₂ have the same derivative at x.

Question 5

What does the order of a differential equation refer to?

Accepted Answer

The highest derivative present in the equation.

Answer

The number of independent variables in the equation.

Answer

The number of dependent variables in the equation.

Answer

The degree of the polynomial on the left-hand side of the equation.

Question 6

Which of the following is NOT a type of higher-order linear homogeneous differential equation?

Accepted Answer

Riccati equation

Answer

Cauchy-Euler equation

Answer

Bessel's equation

Answer

Legendre's equation

Question 7

What can the Laplace transform be used to solve?

Accepted Answer

Initial value problems involving higher-order differential equations.

Answer

Boundary value problems involving partial differential equations.

Answer

Systems of differential equations.

Answer

Partial differential equations involving Fourier series.

Question 8

Determine if the functions y1(x) = e^x and y2(x) = e^(-x) are linearly independent.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 9

For the differential equation y'' - 2y' + y = 0, determine the general solution.

Accepted Answer

y = c₁e^x + c₂e^(-x)

Answer

y = c₁e^(2x) + c₂e^(-2x)

Answer

y = c₁cos(x) + c₂sin(x)

Answer

y = c₁cosh(x) + c₂sinh(x)

Question 10

Define the order of a differential equation.

Accepted Answer

The highest derivative appearing in the equation.

Answer

The number of independent variables in the equation.

Answer

The number of dependent variables in the equation.

Answer

The degree of the highest derivative in the equation.

Question 11

Identify the example of a non-linear differential equation.

Accepted Answer

y' = x^2 + y

Answer

y'' + y' + y = 0

Answer

y' = ky

Answer

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

Question 12

If the Wronskian of a set of functions is nonzero at a point, what can be concluded about the functions at that point?

Accepted Answer

They are linearly independent.

Answer

They are linearly dependent.

Answer

They are linearly dependent on the entire interval.

Answer

They are linearly independent on the entire interval.

Question 13

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

Accepted Answer

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

Answer

y = c1sin(2x) + c2cos(2x)

Answer

y = c1cosh(2x) + c2sinh(2x)

Answer

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

Question 14

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

Accepted Answer

y1y2' - y2y1'

Answer

y1'y2' - y2'y1'

Answer

y1y2'' - y2y1''

Answer

y1'y2'' - y2'y1''

Question 15

Which of the following is a direct consequence of the Wronskian being nonzero for two functions?

Accepted Answer

The functions are linearly independent.

Answer

The differential equation has a unique solution.

Answer

The functions are orthogonal to each other.

Answer

The functions form a complete set of solutions.

Question 16

The method of variation of parameters is primarily used to solve which type of differential equation?

Accepted Answer

Nonhomogeneous linear differential equations with variable coefficients

Answer

Homogeneous linear differential equations

Answer

Nonhomogeneous linear differential equations with constant coefficients

Answer

Systems of differential equations

Question 17

Which of the following is a necessary and sufficient condition for linear independence of a set of functions?

Accepted Answer

Determinant of the Wronskian must be nonzero

Answer

All of the functions in the set must be linearly dependent

Question 18

Consider the differential equation y''' + 2y'' - y' - 2y = 0. What is the order of this differential equation?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 19

A differential equation that has the form y'' + p(x)y' + q(x)y = f(x), where f(x) is a continuous function, is called a:

Accepted Answer

Non-homogeneous differential equation

Answer

Homogeneous differential equation

Answer

Linear differential equation

Question 20

The solution to the initial value problem y'' - 2y' + 2y = 0, y(0) = 1, y'(0) = 0 is:

Accepted Answer

y(x) = e^x cos x

Answer

y(x) = e^x sin x

Answer

y(x) = cos x

Question 21

Identify the linear homogeneous differential equation from the following options:

Accepted Answer

y'' + 2y' + y = 0

Answer

y''' - y' = sin(x)

Answer

y'' + y^2 = 0

Answer

y' = x^2 + y

Question 22

Which condition must hold for a set of functions to be linearly independent?

Accepted Answer

Their Wronskian must be nonzero.

Answer

Their Wronskian must be zero.

Answer

Their derivatives must be linearly independent.

Answer

Their integrals must be linearly independent.

Question 23

Solve the differential equation y''' - 3y'' + 2y' = 0.

Accepted Answer

y = c1e^x + c2e^(2x) + c3x

Answer

y = c1 + c2e^x + c3e^(-x)

Answer

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

Answer

y = c1 + c2e^x

Question 24

Consider the differential equation y'' + p(x)y' + q(x)y = 0. If y1(x) and y2(x) are solutions, which of the following is also a solution?

Accepted Answer

c1y1(x) + c2y2(x)

Answer

y1(x) + y2(x)

Answer

y1(x) - y2(x)

Answer

y1(x)y2(x)

Question 25

What type of singular point is a differential equation said to have if its coefficients are analytic at that point?

Accepted Answer

Regular

Answer

Irregular

Answer

Ordinary

Answer

Extraordinary

Question 26

Identify the type of singular point at x = 0 for the differential equation xy'' + (x-1)y' + y = 0.

Accepted Answer

Regular singular point

Answer

Irregular singular point

Answer

Ordinary point

Answer

Extraordinary point

Question 27

Which technique is used to solve a differential equation with a regular singular point?

Accepted Answer

Method of Frobenius

Answer

Method of undetermined coefficients

Answer

Variation of parameters

Answer

Integrating factor

Question 28

The Wronskian of a set of functions y₁, y₂, ..., yₙ is a determinant that can be used to:

Accepted Answer

Determine whether the functions are linearly independent

Answer

Calculate the eigenvalues

Answer

Solve the differential equation

Answer

Find the general solution

Question 29

Consider the differential equation y''' - 3y'' + 2y' - y = 0. What is the order of this equation?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 30

What is the significance of the discriminant, Δ, in the characteristic equation of a homogeneous linear differential equation?

Accepted Answer

It determines the nature of the solutions.

Answer

It is irrelevant to the solution

Answer

It gives the values of the constants in the general solution.

Answer

It can be used to find particular solutions.

Question 31

The method of variation of parameters can be used to find a particular solution to a nonhomogeneous linear differential equation:

Accepted Answer

If the Wronskian of the fundamental set of solutions is nonzero.

Answer

If the forcing function is a constant.

Answer

If the differential equation is autonomous

Answer

If the differential equation is of order 1.

Question 32

Which of the following is a necessary and sufficient condition for a set of functions to be linearly independent over an interval (a, b)?

Accepted Answer

Their Wronskian is nonzero on (a, b).

Answer

They are all solutions to the same differential equation on (a, b).

Answer

Their derivatives are linearly independent on (a, b).

Answer

They have no common zeros on (a, b).

Question 33

The order of a differential equation refers to:

Accepted Answer

The highest order derivative that appears in the equation.

Answer

The number of independent variables in the equation.

Answer

The number of dependent variables in the equation.

Question 34

Consider the differential equation y^(4) + y'' = 0. The characteristic equation is:

Accepted Answer

r^4 + r^2 = 0

Answer

r^4 - r^2 = 0

Answer

r^2 + r^4 = 0

Question 35

Which of the following is a characteristic of higher-order differential equations?

Accepted Answer

They involve derivatives of order greater than or equal to two.

Answer

They are always linear.

Answer

They have only one solution.

Question 36

The initial conditions for a third-order differential equation specify:

Accepted Answer

The values of the solution and its first two derivatives at a given point.

Answer

The form of the general solution.

Answer

The values of the coefficients in the differential equation.

Question 37

Consider the differential equation y''' + 4y'' + 3y' + 2y = sin(x). Which method is most appropriate to solve for the particular solution?

Accepted Answer

Method of Undetermined Coefficients

Answer

Laplace Transform

Answer

Frobenius Method

Answer

Variation of Parameters

Question 38

Which of the following is a necessary condition for three solutions to be linearly independent?

Accepted Answer

Their Wronskian is not zero at any point in the interval.

Answer

Their derivatives are linearly independent.

Answer

Their graphs do not intersect.

Question 39

Consider the differential equation y'' - 2y' + 5y = 0. What is the characteristic equation?

Accepted Answer

r^2 - 2r + 5 = 0

Answer

r^2 + 2r + 5 = 0

Answer

r^2 - 2r - 5 = 0

Answer

r^2 + 2r - 5 = 0

Question 40

Which of the following series solutions is valid for the differential equation y'' + xy = 0 around the ordinary point x = 0?

Accepted Answer

y = a0 + a1x + a2x^2 + ...

Answer

y = a0 + a1sin(x) + a2cos(x) + ...

Answer

y = a0e^x + a1xe^x + a2x^2e^x + ...

Question 41

Consider the differential equation y''' + p(x)y'' + q(x)y' + r(x)y = 0. If p(x) = x, q(x) = x^2, and r(x) = x^3, what is the order of the equation?

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 42

Identify the second-order linear homogeneous differential equation from the options below.

Accepted Answer

y'' + 3y' + 2y = 0

Answer

y'' + y' + y^2 = 0

Answer

y'' + y^2 = 1

Answer

y' + y = sin(x)

Question 43

What is the formula for the Wronskian of two functions, y1(x) and y2(x)?

Accepted Answer

W(y1, y2) = y1y2' - y1'y2

Answer

W(y1, y2) = y1y2

Answer

W(y1, y2) = y1'y2 - y1y2'

Answer

W(y1, y2) = y1 + y2

Question 44

If the Wronskian of two solutions to a second-order linear homogeneous differential equation is zero, what can be concluded about the solutions?

Accepted Answer

The solutions are linearly dependent.

Answer

The solutions are not solutions to the differential equation.

Answer

The solutions are not necessarily linearly dependent or independent.

Answer

The solutions are linearly independent.

Question 45

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

Accepted Answer

y(x) = c1e^(2x) + c2xe^(2x)

Answer

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

Answer

y(x) = c1e^(-2x) + c2xe^(-2x)

Answer

y(x) = c1e^(x) + c2e^(4x)

Question 46

Which method is commonly used to find a particular solution to the non-homogeneous differential equation y'' + 4y = cos(2x)?

Accepted Answer

Undetermined coefficients

Answer

Separation of variables

Answer

Laplace transforms

Answer

Variation of parameters

Question 47

Determine the order of the differential equation: (d^3y/dx^3) + 2(d^2y/dx^2) - 5(dy/dx) + 3y = 0

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 48

A fourth-order linear homogeneous differential equation has a characteristic equation with roots r = 1, r = 1, r = -2, and r = -2. How many linearly independent solutions can be found for this equation?

Accepted Answer

4

Answer

1

Answer

3

Answer

2

Question 49

Which of the following is NOT a characteristic of a fundamental set of solutions for a homogeneous linear differential equation?

Accepted Answer

The solutions must be linearly dependent.

Answer

The solutions must form a basis for the solution space.

Answer

The solutions must span the solution space.

Answer

The solutions must be linearly independent.

Question 50

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

Accepted Answer

y = (c₁ + c₂x)e^(2x)

Answer

y = c₁x + c₂x^2

Answer

y = c₁e^x + c₂e^(2x)

Answer

y = c₁e^(2x) + c₂e^(-2x)

Question 51

Are the functions y₁(x) = x and y₂(x) = x² linearly independent on the interval (-∞, ∞)?

Accepted Answer

Linearly independent

Answer

None of the above

Answer

Cannot be determined

Answer

Linearly dependent

Question 52

What is the Wronskian of the functions y₁(x) = e^x and y₂(x) = e^(-x)?

Accepted Answer

-2

Answer

2

Answer

0

Answer

1

Question 53

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

Accepted Answer

Third-order

Answer

Second-order

Answer

Fourth-order

Answer

First-order

Question 54

Which of the following represents the general solution to the differential equation y'' + 4y = sin(2x)?

Accepted Answer

y = c₁cos(2x) + c₂sin(2x) + (1/2)xsin(2x) - (1/4)cos(2x)

Answer

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

Answer

y = c₁e^(2x) + c₂e^(-2x) + (1/2)xsin(2x) - (1/4)cos(2x)

Answer

y = c₁cos(2x) + c₂sin(2x)

Question 55

Is the differential equation y' - 2y = x² linear or nonlinear?

Accepted Answer

Linear

Answer

Nonlinear

Answer

None of the above

Answer

Cannot be determined

Question 56

What is the solution to the initial value problem y'' + y = 0, y(0) = 1, y'(0) = 0?

Accepted Answer

y = cos(x)

Answer

y = x

Answer

y = e^x

Answer

y = sin(x)

Question 57

What is the characteristic equation for the differential equation y'' - 9y = 0?

Accepted Answer

r² - 9 = 0

Answer

r + 9 = 0

Answer

r - 9 = 0

Answer

r² + 9 = 0

Question 58

What is the general solution to the differential equation y''' - 6y'' + 11y' - 6y = 0?

Accepted Answer

y = c₁e^x + c₂e^(2x) + c₃e^(3x)

Answer

y = c₁x^2 + c₂x + c₃

Answer

y = c₁e^x + c₂e^(-2x) + c₃e^(-3x)

Answer

y = c₁e^(-x) + c₂e^(-2x) + c₃e^(-3x)

Question 59

Is the differential equation y⁽⁴⁾ + 3y''' + 2y'' + y' - y = 0 homogeneous or nonhomogeneous?

Accepted Answer

Homogeneous

Answer

Nonhomogeneous

Answer

Cannot be determined

Answer

None of the above