Modelling With First Order Equations

Question 1

What is a first order differential equation?

Answer 1

A first order differential equation is an equation that involves the first derivative of a function and the function itself.

Question 2

True or False: A first order equation can be written in the form dy/dx = f(x, y).

Answer 2

True

Question 3

Fill in the blank: The general solution of a first order linear differential equation is of the form y = ______.

Answer 3

Ce^(kx) + particular solution

Question 4

What does the ‘C’ represent in the general solution of a first order differential equation?

Answer 4

The constant of integration.

Question 5

Which of the following is a method for solving first order differential equations? (A) Separation of variables (B) Quadratic formula (C) Completing the square

Answer 5

A

Question 6

True or False: The initial value problem for a first order equation provides a specific solution.

Answer 6

True

Question 7

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

Answer 7

dy/dx + P(x)y = Q(x)

Question 8

Fill in the blank: The integrating factor for the equation dy/dx + P(x)y = Q(x) is given by e^(_________).

Answer 8

∫P(x)dx

Question 9

What type of solution does the method of separation of variables yield?

Answer 9

A general solution involving a constant.

Question 10

True or False: All first order differential equations can be solved using the same method.

Answer 10

False

Question 11

What is the purpose of an integrating factor?

Answer 11

To transform a non-exact equation into an exact one.

Question 12

Fill in the blank: In the context of first order equations, a ‘particular solution’ is a solution that satisfies both the differential equation and the ______.

Answer 12

initial conditions

Question 13

Which of the following represents a separable differential equation? (A) dy/dx = y^2 + x (B) dy/dx = xy (C) dy/dx = e^x

Answer 13

B

Question 14

What is the key feature of a homogeneous first order differential equation?

Answer 14

It can be expressed in the form dy/dx = f(y/x).

Question 15

True or False: The solution to a first order differential equation can be a function that is not continuous.

Answer 15

False

Question 16

What does the term ‘exact equation’ refer to in first order differential equations?

Answer 16

An equation that can be expressed as M(x,y)dx + N(x,y)dy = 0, where ∂M/∂y = ∂N/∂x.

Question 17

Fill in the blank: A first order linear differential equation can be solved using the method of ______.

Answer 17

undetermined coefficients

Question 18

Which equation represents a first order separable equation?

Answer 18

dy/dx = g(x)h(y)

Question 19

True or False: The initial condition can affect the uniqueness of the solution for a first order differential equation.

Answer 19

True

Question 20

What is the typical first step in solving a separable differential equation?

Answer 20

Rearranging the equation to isolate variables.

Question 21

Fill in the blank: The solution to dy/dx = ky, where k is a constant, is of the form y = ______.

Answer 21

Ce^(kt)

Question 22

What is the significance of the slope in a first order differential equation?

Answer 22

It represents the rate of change of the dependent variable with respect to the independent variable.

Question 23

Which of the following is NOT a characteristic of first order differential equations? (A) They involve only first derivatives (B) They can model growth and decay (C) They can only be solved graphically

Answer 23

C

Question 24

What is the method of characteristics used for?

Answer 24

To solve certain types of first order partial differential equations.

Question 25

True or False: A first order differential equation can be linear or nonlinear.

Answer 25

True

Question 26

Fill in the blank: The solution to a first order linear differential equation can often be expressed as a ______.

Answer 26

combination of complementary and particular solutions