Linear First Order Differential Equations

Question 1

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

Answer 1

The general form is dy/dx + P(x)y = Q(x).

Question 2

True or False: The term P(x) in a first-order linear differential equation can be a function of y.

Answer 2

False.

Question 3

Fill in the blank: The solution to a first-order linear differential equation can be found using an _______ factor.

Answer 3

integrating

Question 4

What is the integrating factor for the equation dy/dx + 2y = 3?

Answer 4

e^(2x)

Question 5

Multiple Choice: Which of the following is a first-order linear differential equation? A) y’’ + 3y’ + 2y = 0 B) dy/dx + y = sin(x) C) y’’’ + 4y = x

Answer 5

B) dy/dx + y = sin(x)

Question 6

What is the standard form of the equation dy/dx = 4y + 2?

Answer 6

dy/dx - 4y = 2.

Question 7

True or False: The solution to a first-order linear differential equation is unique given an initial condition.

Answer 7

True.

Question 8

Fill in the blank: The first step in solving a first-order linear differential equation is to find the _______.

Answer 8

integrating factor

Question 9

What is the solution to the equation dy/dx + 3y = 6?

Answer 9

y(x) = 2 - Ce^(-3x).

Question 10

Multiple Choice: In the equation dy/dx + xy = x, what is P(x)? A) x B) 1 C) 0

Answer 10

A) x

Question 11

What is the integrating factor for the equation dy/dx - 5y = 10?

Answer 11

e^(-5x)

Question 12

True or False: The term Q(x) represents the non-homogeneous part of the equation.

Answer 12

True.

Question 13

Which equation represents a first-order linear differential equation?

Answer 13

dy/dx + 7y = 4x.

Question 14

Fill in the blank: The solution to a first-order linear differential equation can often be expressed in terms of _______.

Answer 14

exponentials

Question 15

Multiple Choice: What is the solution to dy/dx + y = 0? A) y = Ce^x B) y = Ce^(-x) C) y = Cx

Answer 15

B) y = Ce^(-x)

Question 16

What does the term ‘linear’ refer to in first-order linear differential equations?

Answer 16

It refers to the fact that the dependent variable and its derivatives appear to the first power.

Question 17

True or False: A first-order linear differential equation can have variable coefficients.

Answer 17

True.

Question 18

What is the standard form of the equation dy/dx + y/x = 3?

Answer 18

dy/dx + (1/x)y = 3.

Question 19

Fill in the blank: The solution to dy/dx + 4y = 8 is y(x) = _______.

Answer 19

2 - Ce^(-4x)

Question 20

Multiple Choice: In the equation dy/dx = 2y + sin(x), which term is the dependent variable? A) dy B) y C) x

Answer 20

B) y

Question 21

What is the solution for the initial value problem dy/dx + y = 1 with y(0) = 0?

Answer 21

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

Question 22

True or False: The general solution of a first-order linear differential equation contains a constant of integration.

Answer 22

True.

Question 23

What is the form of the solution for the equation dy/dx + 3y = 0?

Answer 23

y(x) = Ce^(-3x).

Question 24

Fill in the blank: The equation dy/dx + 1 = 2y can be rewritten as _______.

Answer 24

dy/dx - 2y = -1.