Master Linear Equations: Enhance Your Differential Equations Skills

Question 1

Consider the differential equation: dy/dx - xy = x. Which of the following substitutions can be used to transform this equation into a linear equation?

Accepted Answer

y = u/x

Answer

y = u + x

Answer

y = ux

Answer

y = u - x

Question 2

Which of the following is the particular solution to the differential equation: dy/dx + 2y = e^x, given that y(0) = 1?

Accepted Answer

y = (1/2)e^x + (1/2)e^(-2x)

Answer

y = (1/2)e^x - (1/2)e^(-2x)

Answer

y = (1/2)e^(-x) + (1/2)e^(2x)

Answer

y = (1/2)e^(-x) - (1/2)e^(2x)

Question 3

Which method is appropriate for solving a first-order linear differential equation of the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are non-constant functions?

Accepted Answer

Integrating factor

Answer

Separation of variables

Answer

Substitution

Question 4

What is the standard form of a first-order linear equation?

Accepted Answer

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

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

Question 5

What is the integrating factor for the linear equation (x + 2y) dx + (2x + y) dy = 0?

Accepted Answer

1/(x + y)

Answer

1/x

Answer

1/y

Answer

1/(x - y)

Question 6

What is the particular solution of the linear equation y' + 2y = x, given that y(0) = 1?

Accepted Answer

y = (x - 1) e^(-2x)

Answer

y = (x + 1) e^(-2x)

Answer

y = (x - 1) e^(2x)

Answer

y = (x + 1) e^(2x)

Question 7

Solve the initial value problem y' - y = 2e^(3x), y(0) = 1.

Accepted Answer

y = (2/3)e^(3x) + (1/3)e^x

Answer

y = (1/3)e^(3x) + (2/3)e^x

Answer

y = e^(3x) - e^x

Answer

y = e^x - 2e^(3x)

Question 8

Which of the following is a homogeneous linear equation?

Accepted Answer

y' + xy = 0

Answer

y' + y = x

Answer

y' + 2y = e^x

Answer

y' - y/x = sin(x)

Question 9

To solve the equation y' = y + e^x, the appropriate method is:

Accepted Answer

Integrating factor

Answer

Separation of variables

Answer

Variation of parameters

Answer

Undetermined coefficients

Question 10

To solve the differential equation (1 + e^x)y' + (2 + xe^x)y = 0, what should be used as the integrating factor?

Accepted Answer

e^x

Answer

e^-x

Answer

2e^x

Answer

xe^x

Question 11

For a function to be an integrating factor of a first-order linear differential equation, what condition must it satisfy?

Accepted Answer

Its derivative must be equal to the coefficient of y

Answer

It must be continuous

Answer

It must be differentiable

Answer

Its integral must be equal to the coefficient of y'

Question 12

Solve the differential equation x^2y' + 2xy - 6y = x^3.

Accepted Answer

y = x^2 + C/x^2

Answer

y = x^2 - C/x^2

Answer

y = x + C/x

Answer

y = x - C/x

Question 13

Which technique is appropriate for solving a system of linear differential equations?

Accepted Answer

Matrix methods

Answer

Elimination method

Answer

Integrating factors

Answer

Variation of parameters

Question 14

Find the integrating factor for the differential equation (x + y)dx + (x - y)dy = 0.

Accepted Answer

1/(x^2 - y^2)

Answer

1/(x + y)

Answer

x - y

Answer

x + y

Question 15

How does an integrating factor aid in determining the exactness of a differential equation?

Accepted Answer

It transforms the equation into an exact form, enabling a simple solution by making the left-hand side a total derivative.

Answer

It always makes the equation inexact.

Answer

It is only applicable to linear equations.

Answer

Any arbitrary function can be used as an integrating factor.

Question 16

To solve the equation dy/dx + P(x)y = Q(x) using the integrating factor method when P(x) and Q(x) are not continuous, what modification is necessary?

Accepted Answer

Find an integrating factor u(x) such that u(x)P(x) is continuous.

Answer

Find an integrating factor u(x) such that u(x)Q(x) is continuous.

Answer

The integrating factor method cannot be modified for non-continuous coefficients.

Answer

No modification is needed; the method works the same for continuous and non-continuous coefficients.

Question 17

Identify the linear equation among the following:

Accepted Answer

y = 2x + 5

Answer

y = ln(x)

Answer

y = e^x

Answer

y = x^2 + 2x - 1

Question 18

Determine the slope of the line represented by the equation: y = -3x + 2

Accepted Answer

-3

Answer

-2

Answer

3

Answer

2

Question 19

Identify the homogeneous linear equation among the following:

Accepted Answer

y' + y = 0

Answer

y' + y = x

Answer

y' + 2y = x^2

Answer

y' - y = x

Question 20

Determine whether the linear equation (xy + 2y)dx + (x^2 + 2x)dy = 0 is exact.

Accepted Answer

Exact

Answer

Separable

Answer

Linear

Answer

Not exact

Question 21

Which of the following is an example of a homogeneous linear equation?

Accepted Answer

y' = xy + y

Answer

y' = x^2 + y

Answer

y' = xy - y

Question 22

Find the integrating factor for the linear equation: (x^2 + y^2)dx + 2xydy = 0.

Accepted Answer

1/(x^2 + y^2)

Answer

-1/(x^2 + y^2)

Answer

x^2 + y^2

Question 23

Solve the initial value problem: dy/dx - (2x/y) = 0, y(1) = 2.

Accepted Answer

y = 2e^(x^2)

Answer

y = 2x^2

Answer

y = -2e^(x^2)

Answer

y = -2x^2

Question 24

Identify the type of singularity at x = 0 for the equation y' + (2x/y) = 0.

Accepted Answer

Removable

Answer

Essential

Answer

Regular

Question 25

Determine the type of linear equation: y' - (2xy) = 3.

Accepted Answer

Non-homogeneous linear

Answer

Homogeneous linear

Answer

Bernoulli

Answer

Exact

Question 26

Which of the following methods is used to solve the first-order linear equation y' + P(x)y = Q(x)?

Accepted Answer

Integrating Factor

Answer

Separation of Variables

Answer

Variation of Parameters

Answer

Laplace Transform

Question 27

Find the integrating factor for the linear equation y' - 2y = e^x.

Accepted Answer

e^{-2x}

Answer

e^2x

Answer

-e^{-2x}

Answer

-e^2x

Question 28

A first-order linear differential equation has the form:

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

y^{(n)} + P(x)y^{(n-1)} + Q(x)y^{(n-2)} + ... + R(x)y = 0

Question 29

Solve the linear equation y'' - 2y' + y = e^x.

Accepted Answer

y = Ce^x + x(Ce^x + 1)

Answer

y = Ce^x + x(Ce^x - 1)

Answer

y = Ce^{-x} + x(Ce^{-x} - 1)

Answer

y = Ce^{-x} + x(Ce^{-x} + 1)

Question 30

What is the general form of a linear equation in two variables?

Accepted Answer

Ax + By = C

Answer

y = e^(mx + b)

Answer

y = ln(mx + b)

Answer

y = ax^2 + bx + c

Question 31

In the linear equation y = 2x - 5, what is the slope?

Accepted Answer

2

Answer

-5

Answer

5

Answer

-2

Question 32

Which of the following equations represents a vertical line?

Accepted Answer

x = 3

Answer

y - 2 = 0

Answer

y = 4x + 2

Answer

2x + 3y = 6

Question 33

What is the y-intercept of the linear equation y = -3x + 1?

Accepted Answer

1

Answer

-1

Answer

-3

Answer

0

Question 34

Which of the following characteristics defines a linear function?

Accepted Answer

Constant rate of change (slope)

Answer

Domain restricted to natural numbers

Answer

Asymptotic behavior

Answer

Periodic behavior

Question 35

A first-order differential equation of the form $ dy/dx + P(x) y = Q(x) $ is called a:

Accepted Answer

Linear equation

Answer

Nonlinear equation

Answer

Bernoulli equation

Answer

Homogeneous equation

Question 36

If $ \mu(x) $ is an integrating factor for the differential equation $ dy/dx + P(x) y = Q(x) $, then multiplying both sides of the equation by $ \mu(x) $ transforms it into a(n):

Accepted Answer

Exact equation

Answer

Linear homogeneous equation

Answer

Separable equation

Answer

First-order nonlinear equation

Question 37

Which of the following is NOT a common method for finding an integrating factor for a linear differential equation?

Accepted Answer

Exact equation test

Answer

Comparison with a known equation

Answer

Inspection

Answer

Method of undetermined coefficients

Question 38

The integrating factor for the differential equation $ dy/dx + (2/x)y = x^2 $ is:

Accepted Answer

$ \mu(x) = x^2 $

Answer

$ \mu(x) = e^{2/x} $

Answer

$ \mu(x) = e^{x^2} $

Answer

$ \mu(x) = x $

Question 39

The differential equation $ y' + 2xy = x $ is linear in:

Accepted Answer

$ y $

Answer

$ x $

Answer

$ y' $

Question 40

The general solution to the differential equation $ y' = 2x + y $ is:

Accepted Answer

$ y = Ce^x - x^2 - x $

Answer

$ y = Ce^x + x^2 + x $

Answer

$ y = Ce^{-x} - x^2 - x $

Answer

$ y = Ce^{-x} + x^2 + x $

Question 41

Which of the following methods is commonly used to solve the linear differential equation y' + P(x)y = Q(x)?

Accepted Answer

Integrating factors

Answer

Separation of variables

Answer

Method of undetermined coefficients

Question 42

The integrating factor for the differential equation y' + 2xy = x is:

Accepted Answer

e^(x^2)

Answer

x^2

Answer

1/x^2

Answer

e^(-x^2)

Question 43

Which of the following is NOT a typical step in solving a linear differential equation using integrating factors?

Accepted Answer

Substitute the integrating factor into the original equation.

Answer

Find the integrating factor.

Answer

Multiply both sides of the equation by the integrating factor.

Question 44

The general solution to the linear differential equation y' + (2/x)y = x^2 is:

Accepted Answer

y = x^2/2 + Ce^(-2lnx)

Answer

y = x^2 + Ce^(-lnx)

Answer

y = -x^2 + Ce^(-2lnx)

Answer

y = -x^2/2 + Ce^(-2lnx)

Question 45

In the equation y' + P(x)y = Q(x), if P(x) = 0, the equation is considered:

Accepted Answer

Separable

Answer

Exact

Answer

Homogeneous

Answer

Non-linear

Question 46

A first-order differential equation M(x, y)dx + N(x, y)dy = 0 is exact if and only if:

Accepted Answer

∂M/∂y = ∂N/∂x

Answer

∂M/∂x = ∂N/∂y

Answer

M/N = constant

Answer

M = N

Question 47

Determine the integrating factor for the differential equation (x + y)dx + (x - y)dy = 0.

Accepted Answer

1/(x^2 - y^2)

Answer

x^2 - y^2

Answer

x + y

Answer

1/(x + y)

Question 48

Which of the following is the correct sequence of steps to solve a linear first-order differential equation using an integrating factor?

Accepted Answer

Identify P(x), calculate the integrating factor, multiply both sides of the equation by the integrating factor, rewrite the left side as the derivative of a product, integrate both sides, solve for y.

Answer

Identify P(x), multiply both sides of the equation by the integrating factor, calculate the integrating factor, rewrite the left side as the derivative of a product, integrate both sides, solve for y.

Answer

Identify Q(x), calculate the integrating factor, multiply both sides of the equation by the integrating factor, integrate both sides, solve for y.

Answer

Identify P(x), calculate the integrating factor, multiply only the left side of the equation by the integrating factor, rewrite the left side as the derivative of a product, integrate both sides, solve for y.

Question 49

Find the integrating factor for the linear first-order differential equation dy/dx + 2y = x.

Accepted Answer

e^(2x)

Answer

2x

Answer

x^2

Answer

e^(-2x)

Question 50

What is the general solution of the differential equation dy/dx - y = e^x?

Accepted Answer

y = Ce^x + xe^x

Answer

y = Ce^x - xe^x

Answer

y = Ce^(-x) - xe^x

Answer

y = Ce^(-x) + xe^x

Question 51

Solve the initial value problem: dy/dx + 2y = 4x, with the initial condition y(0) = 1.

Accepted Answer

y = 2x - 1 + 2e^(-2x)

Answer

y = 2x + 1 + 2e^(-2x)

Answer

y = 2x - 1 - 2e^(-2x)

Answer

y = 2x + 1 - 2e^(-2x)

Question 52

Which statement best describes a linear first-order differential equation?

Accepted Answer

The equation is of the first degree in the dependent variable and its derivative.

Answer

The dependent variable and its derivatives appear only in the first power

Answer

The equation can be written in the form dy/dx + P(x)y = Q(x)

Answer

The equation only contains the first derivative of the dependent variable.

Question 53

In the context of linear differential equations, a homogeneous equation is characterized by:

Accepted Answer

The absence of a term solely dependent on the independent variable.

Answer

The absence of a term involving the dependent variable.

Answer

Having constant coefficients for all terms

Answer

Having only one independent variable

Question 54

What is the general solution of the homogeneous linear first-order differential equation dy/dx + 4y = 0?

Accepted Answer

y = Ce^(-4x)

Answer

y = -4x + C

Answer

y = 4x + C

Answer

y = Ce^(4x)

Question 55

A linear first-order differential equation can be solved using the integrating factor method if and only if:

Accepted Answer

The coefficients P(x) and Q(x) are continuous functions on some interval

Answer

The equation has constant coefficients

Answer

The equation is homogeneous