What a differential equation means | CAIE 9709 Pure Mathematics 3 Notes

Chapters

Pure Mathematics 3

Mathematics9709

Start with the meaning of the derivative

Definition

What is meant by differential equation?

Model answer: An equation containing a derivative of an unknown function.

A derivative is a rate at one instant

xyrise ÷ run

dy/dx = instantaneous rate

A differential equation tells us how the rate is connected to the current variables.

In $\dfrac{dy}{dx}$ , $x$ is the independent variable and $y$ is the dependent variable: $y$ is a function of $x$ .
The derivative $\dfrac{dy}{dx}$ is the instantaneous rate at which $y$ changes as $x$ changes.
A first-order differential equation contains a first derivative such as $dy/dx$ , but no second or higher derivative. This syllabus solves first-order equations.

Key idea

The solution of an algebraic equation is usually one or more numbers. The solution of a differential equation is a function or a family of functions.

General solution first, particular solution second

One rate law gives a family of curves

(0, 2)selected solution

DE → family

condition → one curve

The differential equation gives a family; one condition selects one member.

For example, $dy/dx=3x^2$ asks for a function whose derivative is $3x^2$ . Integrating gives $y=x^3+C$ .

A general solution contains an arbitrary constant such as $C$ . It represents every curve in a family that follows the same rate law.
An initial condition, such as $y=2$ when $x=0$ , gives one point on the required curve.
The selected curve is the particular solution.

Key idea

A first-order general solution normally has one arbitrary constant. One suitable condition determines that constant and selects a particular solution.

Recognise when separation is possible

A separable equation can be rearranged so that one side contains only $y$ and $dy$ , while the other contains only $x$ and $dx$ . A common starting form is a product of an x-only factor and a y-only factor:

\frac{dy}{dx}=f(x)g(y)

Provided $g(y)\ne0$ on the branch being solved, this rearranges to:

\frac{1}{g(y)}\,dy=f(x)\,dx

Common mistake

A sum such as

x+y

is not immediately separable. You cannot put its x-term on one side and its y-term on the other as if the derivative were absent.

Your turnIndependent selection check [2 marks]

Which equation is immediately separable: $\dfrac{dy}{dx}=x(1+y^2)$ or $\dfrac{dy}{dx}=x+y$ ? State the separated form.

Show worked answer

The product form is immediately separable:

\frac{1}{1+y^2}\,dy=x\,dx

The sum $x+y$ cannot be split into one x-only factor and one y-only factor, so it is not immediately separable by this method.

(Pure Mathematics 2 & 3 Coursebook, Ch. 10.1)

Start with the meaning of the derivative

Definition

What is meant by differential equation?

Model answer: An equation containing a derivative of an unknown function.

A derivative is a rate at one instant

xyrise ÷ run

dy/dx = instantaneous rate

A differential equation tells us how the rate is connected to the current variables.

In $\dfrac{dy}{dx}$ , $x$ is the independent variable and $y$ is the dependent variable: $y$ is a function of $x$ .
The derivative $\dfrac{dy}{dx}$ is the instantaneous rate at which $y$ changes as $x$ changes.
A first-order differential equation contains a first derivative such as $dy/dx$ , but no second or higher derivative. This syllabus solves first-order equations.

Key idea

The solution of an algebraic equation is usually one or more numbers. The solution of a differential equation is a function or a family of functions.

General solution first, particular solution second

One rate law gives a family of curves

(0, 2)selected solution

DE → family

condition → one curve

The differential equation gives a family; one condition selects one member.

For example, $dy/dx=3x^2$ asks for a function whose derivative is $3x^2$ . Integrating gives $y=x^3+C$ .

A general solution contains an arbitrary constant such as $C$ . It represents every curve in a family that follows the same rate law.
An initial condition, such as $y=2$ when $x=0$ , gives one point on the required curve.
The selected curve is the particular solution.

Key idea

A first-order general solution normally has one arbitrary constant. One suitable condition determines that constant and selects a particular solution.

Recognise when separation is possible

\frac{dy}{dx}=f(x)g(y)

Provided $g(y)\ne0$ on the branch being solved, this rearranges to:

\frac{1}{g(y)}\,dy=f(x)\,dx

Common mistake

A sum such as

x+y

is not immediately separable. You cannot put its x-term on one side and its y-term on the other as if the derivative were absent.

Your turnIndependent selection check [2 marks]

Which equation is immediately separable: $\dfrac{dy}{dx}=x(1+y^2)$ or $\dfrac{dy}{dx}=x+y$ ? State the separated form.

Show worked answer

The product form is immediately separable:

\frac{1}{1+y^2}\,dy=x\,dx

The sum $x+y$ cannot be split into one x-only factor and one y-only factor, so it is not immediately separable by this method.

(Pure Mathematics 2 & 3 Coursebook, Ch. 10.1)