Do I need a constant of integration `+c` when finding the integrating factor?

No. While technically `\int P(x) dx` has a constant, `e^{P(x)dx + c} = e^{\int P(x)dx} \cdot e^c`. This `e^c` is just another constant multiplier that would apply to the whole equation and eventually get absorbed into the final constant of integration anyway. So, for simplicity, we omit it when calculating the I.F.

What's the difference between a general and a particular solution?

A general solution has the constant `+c` and represents an entire family of functions that satisfy the differential equation. A particular solution is a single, unique function from that family, found by using a specific point (a boundary condition) to determine the exact value of `c`.

How do I deal with `ln|f(y)|` when solving for `y`?

After integrating to get `ln|f(y)| = g(x) + c`, you exponentiate both sides: `|f(y)| = e^{g(x)+c} = e^{g(x)}e^c`. We replace the constant `e^c` with a new constant, say `A`, so `|f(y)| = Ae^{g(x)}`. This means `f(y) = \pm Ae^{g(x)}`. You can absorb the `\pm` into the constant `A`. The sign of `A` is determined by the initial conditions.

My answer for a modelling question is `y = 5e^{kt}` but the mark scheme has `y = Ae^{kt}`. Why?

This often happens when the initial value is used at different stages. If you solve for the constant of integration `c` early, you might get `ln(y) = kt + ln(5)`, which gives `y = e^{kt + ln(5)} = e^{kt}e^{ln(5)} = 5e^{kt}`. If you write the general solution as `y = Ae^{kt}` and then substitute `t=0, y=5`, you get `5 = Ae^0`, so `A=5`. Both methods are correct and lead to the same result.

Can I always use separation of variables?

No, it only works if the equation can be written in the form `dy/dx = f(x)g(y)`. For linear equations like `dy/dx + 2y = x`, you cannot separate the `2y` term from the `x` term, so you must use the integrating factor method.

Where can I learn Differential equations for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 3.8 (Differential equations) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/3-8-differential-equations.

Differential equations 9709 — Free A-Level…

Differential equations

Welcome to the world of differential equations! These are not your typical algebraic equations. Instead of solving for a number, we are solving for a function. Differential equations are fundamental to science and engineering, describing everything from population growth to the cooling of a cup of tea. In this chapter, we'll master the two key techniques required for your P3 exam: separation of variables and the integrating factor method.

BY THE END, YOU CAN…

1Solve first-order differential equations where variables are separable.
2Solve first-order linear differential equations of the form dy/dx + P(x)y = Q(x) using an integrating factor.
3Formulate simple differential equations from problems in context.
4Find a particular solution by using a given boundary condition.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Equations of Change

Differential equations describe how a quantity changes. We learn techniques to 'solve' these equations, which means finding the original function that the rate of change describes.

THINK OF IT LIKE…

Imagine you have a GPS tracker that only tells you your speed and direction at every single moment (the derivative). Solving a differential equation is like using that continuous stream of data to reconstruct the exact path you travelled (the original function). An initial condition, like knowing your starting point, helps you pinpoint the one specific path you took.

1
First order: dy/dx = f(x,y) — slope field shows direction at each point.
2
Separable: ∫(1/g(y)) dy = ∫f(x) dx + c.
3
Linear integrating factor e^∫P dx for dy/dx + Py = Q.
4
Modelling: exponential growth/decay, Newton cooling, logistic.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · First order: dy/dx = f(x,y)

Loading simulation…

Fetching from GeoGebra…

GeoGebra Slope Field Plotter

Read slope at (x,y) — sketch solution through initial condition.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

First order: dy/dx = f(x,y)

First order: dy/dx = f(x,y) — slope field shows direction at each point.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

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

Tap a symbol — great for exam definitions

$\frac{dy}{dx} + P(x)y = Q(x)$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

1. Separable First-Order Differential Equations

The simplest type of differential equation we'll solve is one where the variables can be separated. If you can rearrange an equation so that all the $y$ and dy terms are on one side, and all the $x$ and dx terms are on the other, it's separable. The goal is to get it into a form that can be integrated.

If $\frac{dy}{dx} = f(x)g(y)$ , we can separate it to $\frac{1}{g(y)} dy = f(x) dx$ . Then we integrate both sides: $\int \frac{1}{g(y)} dy = \int f(x) dx + c$ .

Always group dy with $y$ terms and dx with $x$ terms.
Add the constant of integration, + c, immediately after integrating. It's best practice to add it to the side with the independent variable (usually $x$ ).
If given boundary conditions, substitute them in to find the value of $c$ for the particular solution.
Remember that $\int \frac{1}{y} dy = \ln|y| + c$ . The modulus sign is important.

2. Linear First-Order Differential Equations

Not all differential equations are separable. A powerful method for another class of equations involves an 'integrating factor'. This method applies to linear first-order equations, which must first be arranged into a standard form. The magic of the integrating factor is that it transforms one side of the equation into the result of the product rule for differentiation, making it easy to integrate.

Standard Form: $\frac{dy}{dx} + P(x)y = Q(x)$ Integrating Factor (I.F.): $I(x) = e^{\int P(x) dx}$ Resulting Equation: $\frac{d}{dx}(y \cdot I(x)) = Q(x) \cdot I(x)$

EXAM TIP

Before finding the integrating factor, always ensure your equation is in the standard form $dy/dx + P(x)y = Q(x)$ . If you have, for example, $x \frac{dy}{dx} + 2y = x^2$ , you must first divide the entire equation by $x$ to make the coefficient of dy/dx equal to 1.

3. Modelling with Differential Equations

The true power of differential equations lies in their ability to model real-world phenomena. You will often be asked to translate a written description of a rate of change into a mathematical equation. For example, a statement like "the rate of cooling of a body is proportional to the excess temperature over the surroundings" is a classic setup for a differential equation. The key is to correctly identify the variables and the relationship between the rate and the quantities themselves.

Direct Proportionality: "Rate of change of $y$ is proportional to $y$ " means $dy/dt = ky$ .
Inverse Proportionality: "Rate of change of $y$ is inversely proportional to $y$ " means $dy/dt = k/y$ .
Newton's Law of Cooling: If $\theta$ is temperature and $\theta_S$ is surrounding temperature, $d\theta/dt = -k(\theta - \theta_S)$ . The negative sign indicates cooling.
Pay close attention to whether a rate is an 'increase' (positive constant) or 'decrease' (negative constant).

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A differential equation is given by $\frac{dy}{dx} = \frac{y+1}{x^2}$ for $x > 0$ . Given that $y=1$ when $x=1$ , find the particular solution for $y$ in terms of $x$ .

1
This equation is in the form $\frac{dy}{dx} = f(x)g(y)$ , so we can separate the variables.

EXAMPLE 2

A biologist is modelling the number of fish, $N$ , in a lake. The rate of increase of the number of fish is modelled by the differential equation $\frac{dN}{dt} + 0.2N = 100e^{-0.1t}$ , where $t$ is the time in years. Initially ( $t=0$ ), there are 500 fish. Find an expression for $N$ in terms of $t$ .

1
The equation is in the linear form $dN/dt + P(t)N = Q(t)$ with $P(t) = 0.2$ and $Q(t) = 100e^{-0.1t}$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEADifferential equations

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a differential equation?

TAP OR SPACE TO FLIP

ANSWER

An equation that contains a derivative, such as dy/dx, linking a function with one or more of its derivatives.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Always group dy with $y$ terms and dx with $x$ terms.
✓
Add the constant of integration, + c, immediately after integrating. It's best practice to add it to the side with the independent variable (usually $x$ ).
✓
If given boundary conditions, substitute them in to find the value of $c$ for the particular solution.
✓
Remember that $\int \frac{1}{y} dy = \ln|y| + c$ . The modulus sign is important.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Differential Equations

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Differential Equations on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS3.7 Vectors NEXT →3.9 Complex numbers

Differential equations

BY THE END, YOU CAN…

1Solve first-order differential equations where variables are separable.
2Solve first-order linear differential equations of the form dy/dx + P(x)y = Q(x) using an integrating factor.
3Formulate simple differential equations from problems in context.
4Find a particular solution by using a given boundary condition.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Equations of Change

Differential equations describe how a quantity changes. We learn techniques to 'solve' these equations, which means finding the original function that the rate of change describes.

THINK OF IT LIKE…

1
First order: dy/dx = f(x,y) — slope field shows direction at each point.
2
Separable: ∫(1/g(y)) dy = ∫f(x) dx + c.
3
Linear integrating factor e^∫P dx for dy/dx + Py = Q.
4
Modelling: exponential growth/decay, Newton cooling, logistic.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · First order: dy/dx = f(x,y)

Loading simulation…

Fetching from GeoGebra…

GeoGebra Slope Field Plotter

Read slope at (x,y) — sketch solution through initial condition.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step-synced diagram — highlights what to look for in the simulation above.

First order: dy/dx = f(x,y)

First order: dy/dx = f(x,y) — slope field shows direction at each point.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

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

Tap a symbol — great for exam definitions

$\frac{dy}{dx} + P(x)y = Q(x)$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

1. Separable First-Order Differential Equations

If $\frac{dy}{dx} = f(x)g(y)$ , we can separate it to $\frac{1}{g(y)} dy = f(x) dx$ . Then we integrate both sides: $\int \frac{1}{g(y)} dy = \int f(x) dx + c$ .

Always group dy with $y$ terms and dx with $x$ terms.
Add the constant of integration, + c, immediately after integrating. It's best practice to add it to the side with the independent variable (usually $x$ ).
If given boundary conditions, substitute them in to find the value of $c$ for the particular solution.
Remember that $\int \frac{1}{y} dy = \ln|y| + c$ . The modulus sign is important.

2. Linear First-Order Differential Equations

Standard Form: $\frac{dy}{dx} + P(x)y = Q(x)$ Integrating Factor (I.F.): $I(x) = e^{\int P(x) dx}$ Resulting Equation: $\frac{d}{dx}(y \cdot I(x)) = Q(x) \cdot I(x)$

EXAM TIP

3. Modelling with Differential Equations

Direct Proportionality: "Rate of change of $y$ is proportional to $y$ " means $dy/dt = ky$ .
Inverse Proportionality: "Rate of change of $y$ is inversely proportional to $y$ " means $dy/dt = k/y$ .
Newton's Law of Cooling: If $\theta$ is temperature and $\theta_S$ is surrounding temperature, $d\theta/dt = -k(\theta - \theta_S)$ . The negative sign indicates cooling.
Pay close attention to whether a rate is an 'increase' (positive constant) or 'decrease' (negative constant).

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A differential equation is given by $\frac{dy}{dx} = \frac{y+1}{x^2}$ for $x > 0$ . Given that $y=1$ when $x=1$ , find the particular solution for $y$ in terms of $x$ .

1
This equation is in the form $\frac{dy}{dx} = f(x)g(y)$ , so we can separate the variables.

EXAMPLE 2

1
The equation is in the linear form $dN/dt + P(t)N = Q(t)$ with $P(t) = 0.2$ and $Q(t) = 100e^{-0.1t}$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEADifferential equations

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a differential equation?

TAP OR SPACE TO FLIP

ANSWER

An equation that contains a derivative, such as dy/dx, linking a function with one or more of its derivatives.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Always group dy with $y$ terms and dx with $x$ terms.
✓
Add the constant of integration, + c, immediately after integrating. It's best practice to add it to the side with the independent variable (usually $x$ ).
✓
If given boundary conditions, substitute them in to find the value of $c$ for the particular solution.
✓
Remember that $\int \frac{1}{y} dy = \ln|y| + c$ . The modulus sign is important.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Differential Equations

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Differential Equations on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS3.7 Vectors NEXT →3.9 Complex numbers

Revise Differential equations for Cambridge 9709 (3.8)

In simple terms

Explore the concept

Key formulas

Full topic notes

1. Separable First-Order Differential Equations

2. Linear First-Order Differential Equations

3. Modelling with Differential Equations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Differential Equations

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

1. Separable First-Order Differential Equations

2. Linear First-Order Differential Equations

3. Modelling with Differential Equations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Differential Equations

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Differential equations