UK K12 GCSE/A-Level: Year 13 A-Level Upper Sixth Mathematics - Pure Differential Equations, Separable and General

Learning Objectives

By the end of this topic, students will be able to:

• Define and identify separable differential equations and their general solutions
• Apply the method of separation of variables to solve separable differential equations
• Recognize and solve general differential equations using the integrating factor method
• Analyze and interpret the results of solving differential equations, including the identification of equilibrium points and stability
• Apply mathematical modeling techniques to real-world problems involving differential equations

Core Concepts

A differential equation is a mathematical equation involving an unknown function and its derivatives. Separable differential equations are a type of differential equation that can be written in the form:

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

where f(x) and g(y) are functions of x and y, respectively. The general solution of a separable differential equation can be found by integrating both sides of the equation with respect to x and y.

The method of separation of variables involves separating the variables x and y on opposite sides of the equation, and then integrating both sides. This can be done by multiplying both sides of the equation by dx and dy, and then integrating with respect to x and y.

A general differential equation can be written in the form:

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

where P(x) and Q(x) are functions of x. The integrating factor method involves multiplying both sides of the equation by a function of x, called the integrating factor, which is defined as:

?(x) = e^(?P(x)dx)

The integrating factor method involves multiplying both sides of the equation by ?(x), and then integrating both sides with respect to x.

Worked Examples

Example 1: Separable Differential Equation

Solve the differential equation:

dy/dx = 2x / (y + 1)

Using the method of separation of variables, we can separate the variables x and y on opposite sides of the equation:

y + 1 dy/dx = 2x

Integrating both sides with respect to x, we get:

?(y + 1) dy = ?2x dx

Evaluating the integrals, we get:

(y^2 + y) = x^2 + C

where C is a constant.

Example 2: General Differential Equation

Solve the differential equation:

dy/dx + 2y = 3x

Using the integrating factor method, we can multiply both sides of the equation by the integrating factor ?(x) = e^(?2x dx):

?(x) = e^(x^2)

Multiplying both sides of the equation by ?(x), we get:

e^(x^2) dy/dx + 2e^(x^2) y = 3xe^(x^2)

Integrating both sides with respect to x, we get:

?e^(x^2) dy = ?3xe^(x^2) dx

Evaluating the integrals, we get:

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

where C is a constant.

Common Misconceptions

• Students may confuse the method of separation of variables with the integrating factor method.
• Students may not recognize that a differential equation can be separable even if it is not in the standard form.
• Students may not understand the concept of an integrating factor and how it is used to solve general differential equations.

Exam Tips

• Make sure to read the question carefully and identify the type of differential equation being asked.
• Use the method of separation of variables for separable differential equations, and the integrating factor method for general differential equations.
• Make sure to check your solution for any errors or inconsistencies.
• Use mathematical modeling techniques to interpret the results of solving differential equations.

MCQs with Explanations

MCQ 1: [F]

What is the general solution of the differential equation:

dy/dx = 2x / (y + 1)

A) y^2 + y = x^2 + C B) y^2 - y = x^2 + C C) y^2 + y = x^2 - C D) y^2 - y = x^2 - C

Correct answer: A) y^2 + y = x^2 + C

Why the distractors fail:

• B) y^2 - y = x^2 + C is incorrect because the correct solution involves adding y, not subtracting it.
• C) y^2 + y = x^2 - C is incorrect because the correct solution involves adding C, not subtracting it.
• D) y^2 - y = x^2 - C is incorrect because the correct solution involves adding y, not subtracting it.

MCQ 2: [H]

What is the integrating factor for the differential equation:

dy/dx + 2y = 3x

A) e^(?2x dx) B) e^(?2y dy) C) e^(?3x dx) D) e^(?2y dx)

Correct answer: A) e^(?2x dx)

Why the distractors fail:

• B) e^(?2y dy) is incorrect because the integrating factor involves integrating with respect to x, not y.
• C) e^(?3x dx) is incorrect because the integrating factor involves integrating with respect to x, not 3x.
• D) e^(?2y dx) is incorrect because the integrating factor involves integrating with respect to x, not y.

MCQ 3: [F]

What is the general solution of the differential equation:

dy/dx = 2x / (y + 1)

A) y = 2x + C B) y = x^2 + C C) y^2 + y = x^2 + C D) y^2 - y = x^2 + C

Correct answer: C) y^2 + y = x^2 + C

Why the distractors fail:

• A) y = 2x + C is incorrect because the correct solution involves y^2 + y, not just y.
• B) y = x^2 + C is incorrect because the correct solution involves y^2 + y, not just y.
• D) y^2 - y = x^2 + C is incorrect because the correct solution involves adding y, not subtracting it.

MCQ 4: [H]

What is the integrating factor for the differential equation:

dy/dx + 2y = 3x

A) e^(?2x dx) B) e^(?2y dy) C) e^(?3x dx) D) e^(?2y dx)

Correct answer: A) e^(?2x dx)

Why the distractors fail:

• B) e^(?2y dy) is incorrect because the integrating factor involves integrating with respect to x, not y.
• C) e^(?3x dx) is incorrect because the integrating factor involves integrating with respect to x, not 3x.
• D) e^(?2y dx) is incorrect because the integrating factor involves integrating with respect to x, not y.

MCQ 5: [F]

What is the general solution of the differential equation:

dy/dx = 2x / (y + 1)

A) y^2 + y = x^2 + C B) y^2 - y = x^2 + C C) y^2 + y = x^2 - C D) y^2 - y = x^2 - C

Correct answer: A) y^2 + y = x^2 + C

Why the distractors fail:

• B) y^2 - y = x^2 + C is incorrect because the correct solution involves adding y, not subtracting it.
• C) y^2 + y = x^2 - C is incorrect because the correct solution involves adding C, not subtracting it.
• D) y^2 - y = x^2 - C is incorrect because the correct solution involves adding y, not subtracting it.

Short-answer Questions

Question 1

Solve the differential equation:

dy/dx = 2x / (y + 1)

Using the method of separation of variables, find the general solution of the differential equation.

Question 2

Solve the differential equation:

dy/dx + 2y = 3x

Using the integrating factor method, find the general solution of the differential equation.

Question 3

A population of rabbits is growing at a rate proportional to the square root of the population size. If the initial population size is 100, and the population grows to 200 in 2 years, find the population size after 5 years.

Question 4

A company produces a product at a rate that is proportional to the square of the number of machines in use. If the company has 5 machines in use and produces 100 units in the first hour, find the number of units produced in the next hour if the company increases the number of machines to 10.

Question 5

A chemical reaction occurs at a rate that is proportional to the product of the concentrations of two reactants. If the initial concentrations of the two reactants are 1 and 2, and the reaction rate is 0.5, find the concentrations of the two reactants after 2 hours if the reaction rate is 0.2.