First-Order Differential Equations: Interactive Learning Quizzes

Question 1

To solve the differential equation y' + (x^2)y = e^(x^3) using the method of integrating factors, which of the following is the integrating factor?

Accepted Answer

1/(x^2)

Answer

x^2

Answer

e^(x^3)

Answer

1/x

Question 2

An initial value problem (IVP) for a first-order ODE is:

Accepted Answer

An ODE with a specified value of the dependent variable y at a given point (x0, y0)

Answer

An ODE that can be solved using the method of separation of variables

Answer

An ODE that has a unique solution for any initial condition

Answer

An ODE that is linear in the dependent variable y

Question 3

Under what conditions are existence and uniqueness of the solution to an IVP guaranteed?

Accepted Answer

If the functions M and N in the standard form are continuous in a region containing the initial point (x0, y0)

Answer

If the derivative of the dependent variable y exists

Answer

If the initial condition (x0, y0) satisfies the ODE

Answer

If the ODE is linear in the dependent variable y

Question 4

For the differential equation y' + P(x)y = Q(x), the integrating factor is:

Accepted Answer

e^(∫P(x) dx)

Answer

-e^(∫P(x) dx)

Answer

1/e^(∫P(x) dx)

Answer

∫P(x) dx

Question 5

Which of the following is a homogeneous linear ODE?

Accepted Answer

y'' + y = 0

Answer

y' + 2y = x

Answer

y' - xy = 1

Answer

y' + y^2 = e^x

Question 6

Which of the following situations does NOT require the use of a first-order differential equation?

Accepted Answer

Calculating the distance traveled by a falling object

Answer

Predicting the growth of a bacterial colony

Answer

Determining the cooling rate of an object

Question 7

An initial value problem for a first-order differential equation consists of the equation and which additional condition?

Accepted Answer

A value of the dependent variable at a specific point

Answer

A value of the independent variable at a specific point

Answer

A value of the derivative at a specific point

Answer

Values of both the dependent and independent variables at a specific point

Question 8

Which of the following methods can be used to solve a separable differential equation?

Accepted Answer

Separation of variables

Answer

Integration

Answer

Substitution

Answer

Linearization

Question 9

Solve the differential equation dy/dx = (x + y)/(x - y).

Accepted Answer

y = (x + C)/(x - C)

Answer

y = x + C

Answer

y = -x + C

Answer

y = (x - C)/(x + C)

Question 10

Find the particular solution to the initial value problem y' + y = e^x, y(0) = 1.

Accepted Answer

y = e^x + 1

Answer

y = e^x

Answer

y = -e^x

Answer

y = -e^x + 1

Question 11

Which of the following differential equations has a singularity at x = 0?

Accepted Answer

y' = 1/(xy)

Answer

y' = x^2y

Answer

y' = e^xy

Answer

y' = sin(xy)

Question 12

Which method can be used to solve the differential equation y' = (x + y)/x?

Accepted Answer

Integrating factor

Answer

Separation of variables

Answer

Exact equations

Answer

Homogeneous equations

Question 13

The solution to the differential equation y' = -y is

Accepted Answer

y = Ce^-x

Answer

y = Ce^x

Answer

y = -Ce^-x

Answer

y = -Ce^x

Question 14

The equation y' = y^2 + 1 is

Accepted Answer

Neither separable nor exact

Answer

Separable

Answer

Exact

Answer

Both separable and exact

Question 15

The equation y'' - 2y' + 5y = 0 is

Accepted Answer

A homogeneous linear differential equation with constant coefficients

Answer

A non-homogeneous linear differential equation with constant coefficients

Answer

A non-linear differential equation

Answer

A partial differential equation

Question 16

Consider the first-order differential equation $y' = -x$. What is the geometrical interpretation of its solution curves?

Accepted Answer

Curves perpendicular to the line $y = -x$

Answer

Parallel lines with slope -1

Answer

Circles centered at the origin

Question 17

Which of the following is NOT a numerical method for solving first-order differential equations?

Accepted Answer

Integration by parts

Answer

Euler's method

Answer

Runge-Kutta method

Question 18

Which of the following is the definition of a first-order differential equation?

Accepted Answer

An equation involving the first derivative of a dependent variable with respect to an independent variable.

Answer

An equation involving the integrals of a dependent variable with respect to one or more independent variables.

Answer

An equation involving the partial derivatives of a dependent variable with respect to two or more independent variables.

Answer

An equation involving the Laplace transform of a dependent variable.

Question 19

Consider the initial value problem y' = y-1, y(0) = 2. What is the solution to this problem?

Accepted Answer

y = 2e^x - 1

Answer

y = e^x + 1

Answer

y = -e^x + 2

Answer

y = e^(-x) + 2

Question 20

Which of the following is a method for solving first-order differential equations that can be separated?

Accepted Answer

Method of Separation of Variables

Answer

Method of Integrating Factors

Answer

Method of Variation of Parameters

Answer

Method of Undetermined Coefficients

Question 21

Consider the differential equation y' = x^2 + y^2. What type of first-order differential equation is this?

Accepted Answer

Nonlinear

Answer

Linear

Answer

Homogeneous

Answer

Separable

Question 22

What is the general solution to the differential equation y' = ky, where k is a constant?

Accepted Answer

y = Ce^(kx)

Answer

y = C + kx

Answer

y = kCe^(x)

Answer

y = Cx^(ke)

Question 23

For the differential equation (1 + ?(?)^2)dx + (? - ?(?))dy = 0, what is the integrating factor?

Accepted Answer

?^(? - ?(?))

Answer

1 + ?(?)^2

Answer

? - ?(?)

Answer

?^(1 + ?(?)^2)

Question 24

Which theorem guarantees the existence and uniqueness of solutions to first-order differential equations?

Accepted Answer

Picard-Lindelöf Theorem

Answer

Intermediate Value Theorem

Answer

Chain Rule Theorem

Answer

Fundamental Theorem of Calculus

Question 25

For the differential equation ?′(?) = ?(?)^2 - 1, at which points does this equation have singular solutions?

Accepted Answer

?(?) = -1, ?(?) = 1

Answer

?(?) = 0

Answer

?(?) = 1

Answer

?(?) = -1

Question 26

What is the general solution to the differential equation ?′(?) + ?(?) = ?^??

Accepted Answer

?(?) = ?^? + ?

Answer

?(?) = ??^? + ?

Answer

?(?) = ?^(-?) + ?

Answer

?(?) = ??^(-?) + ?

Question 27

Which of the following is NOT a type of first-order differential equation?

Accepted Answer

Partial

Answer

Linear

Answer

Nonlinear

Answer

Homogeneous

Question 28

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

Accepted Answer

y' + P(x)y = Q(x)

Answer

y'' + P(x)y' + Q(x)y = 0

Answer

y''' + P(x)y'' + Q(x)y' + R(x)y = 0

Answer

None of the mentioned

Question 29

For the differential equation y' + P(x)y = Q(x), what is the integrating factor?

Accepted Answer

e^(∫P(x)dx)

Answer

e^(−∫P(x)dx)

Answer

∫P(x)dx

Answer

None of the options provided

Question 30

Which of the following is not a classification of first-order differential equations?

Accepted Answer

Second-Order

Answer

Linear

Answer

Nonlinear

Answer

Homogeneous

Question 31

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

Accepted Answer

y' = f(x, y)

Answer

y = x^2 + y^2

Answer

sin(y) + cos(x) = 0

Answer

y'' + y' = f(x)

Question 32

An initial value problem for a first-order differential equation is:

Accepted Answer

A differential equation with a specified initial value for the dependent variable.

Answer

A system of two or more first-order differential equations.

Answer

A differential equation with no initial conditions.

Question 33

Solve the first-order differential equation y' = xy using the method of separation of variables:

Accepted Answer

y = Ce^(x^2/2)

Answer

y = e^x

Answer

y = x^2

Answer

y = sin(x)

Question 34

An application of first-order differential equations in real life is:

Accepted Answer

Modeling population growth

Answer

Finding the area under a curve

Answer

Calculating the volume of a sphere

Answer

Solving trigonometric identities

Question 35

What is the order of the differential equation d^3y/dx^3 + 2d^2y/dx^2 - 5dy/dx + 6y = 0?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 36

Which of the following is a type of singular solution to a first-order differential equation?

Accepted Answer

Envelope

Answer

Characteristic curve

Answer

Orthogonal trajectory

Answer

Asymptote

Question 37

Identify the first-order differential equation:

Accepted Answer

dy/dx + 2y = x^2

Answer

y'' + 3y' + 2y = sin(x)

Answer

d^2y/dx^2 + y = 0

Question 38

A solution to the differential equation y' = -y is:

Accepted Answer

y = Ce^(-x)

Answer

y = Cx

Answer

y = C

Answer

y = Ce^(x)

Question 39

Which method can be used to solve the differential equation y' = f(x,y) numerically?

Accepted Answer

Euler's method

Answer

Variation of parameters

Answer

Separation of variables

Answer

Integrating factors

Question 40

A first-order differential equation is called homogeneous if it can be expressed in the form:

Accepted Answer

dy/dx = f(y/x)

Answer

dy/dx = f(y)

Answer

dy/dx = f(x)

Question 41

Which of the following differential equations is a non-linear first-order equation?

Accepted Answer

y' + y² = x

Answer

y' + y = sin(x)

Answer

y' + 2y = x

Answer

y' + xy = 0

Question 42

Which of the following equations is NOT a first-order differential equation?

Accepted Answer

d²y/dx² + y = x

Answer

dy/dt = ky(1 - y/N)

Answer

y' = sin(x)y

Answer

dy/dx = y + x

Question 43

An initial value problem (IVP) for a first-order differential equation consists of:

Accepted Answer

The differential equation and the value of the solution at a specific point.

Answer

Two or more differential equations.

Answer

The differential equation and its general solution.

Question 44

The existence and uniqueness theorem for first-order differential equations guarantees a unique solution under specific conditions. Which of the following is NOT a requirement for this theorem?

Accepted Answer

The differential equation is a second-order equation.

Answer

The function defining the differential equation is continuous.

Answer

The function defining the differential equation satisfies a Lipschitz condition.

Question 45

Which of the following equations is a first-order linear differential equation?

Accepted Answer

dy/dx + 2y = x²

Answer

dy/dx = y² + x

Answer

dy/dx = y/x

Answer

dy/dx = sin(xy)

Question 46

The general solution to the differential equation dy/dx = 2y is:

Accepted Answer

y = Ce^(2x)

Answer

y = 2x + C

Answer

y = Ce^(-2x)

Answer

y = 2Ce^(x)

Question 47

The integrating factor for the differential equation dy/dx + 2xy = x is:

Accepted Answer

e^(x²)

Answer

e^(-x²)

Answer

2x

Question 48

Which of these methods is NOT commonly used to solve first-order differential equations?

Accepted Answer

Laplace transforms

Answer

Integrating factors

Answer

Separation of variables

Question 49

A differential equation is considered autonomous if:

Accepted Answer

The independent variable does not appear explicitly in the equation.

Answer

The equation is linear.

Answer

The dependent variable does not appear explicitly in the equation.

Question 50

A population model with limited carrying capacity can be described by a differential equation of the form:

Accepted Answer

dP/dt = kP(1 - P/N)

Answer

dP/dt = kP

Answer

dP/dt = kP²

Answer

dP/dt = k(1 - P/N)