Differential equations, explained with examples

By The STEM-Tutor team · Updated October 2026

A differential equation links a quantity to its rate of change. They model population growth, cooling, radioactive decay and much more, and they appear in the second year of A level Maths.

What a differential equation is

A differential equation contains a derivative, such as dy/dx or dP/dt. Instead of telling you y directly, it tells you how y changes. Solving it means finding the function y that makes the equation true. For example, dy/dx = 2x is solved by y = x² + c, because differentiating x² + c gives 2x.

The general solution contains an arbitrary constant c and describes a whole family of curves. A particular solution uses a given condition, such as y = 3 when x = 0, to find the value of c and pick out one curve from the family.

Separating variables

At A level, the main technique is separation of variables. It works when the equation can be written as dy/dx = f(x)g(y): a function of x times a function of y.

1. Rearrange so all the y terms (with dy) are on one side and all the x terms (with dx) on the other: (1/g(y)) dy = f(x) dx.
2. Integrate both sides. Add one constant of integration, on the x side.
3. Rearrange to make y the subject if the question asks for it.
4. Use any given condition to find the constant.

Worked examples

Example 1: a simple separation

Solve dy/dx = 6x²/y, given y = 4 when x = 0.

• Separate: y dy = 6x² dx.
• Integrate: y²/2 = 2x³ + c.
• At x = 0, y = 4: 8 = c. So y² = 4x³ + 16, and y = √(4x³ + 16), taking the positive root because y = 4 > 0.

Example 2: exponential growth

Solve dy/dx = 3y, given y = 2 when x = 0.

• Separate: (1/y) dy = 3 dx. Integrate: ln|y| = 3x + c.
• Exponentiate: y = e^{3x + c} = Ae^3x, where A = e^c.
• y = 2 at x = 0 gives A = 2. y = 2e^3x.

Example 3: a product on the right

Solve dy/dx = xy, given y = 1 when x = 0.

• (1/y) dy = x dx, so ln|y| = x²/2 + c.
• y = Ae^x²/2, and y(0) = 1 gives A = 1. y = e^x²/2.

Example 4: separating with partial fractions

Solve dy/dx = y/(x(x + 1)) for x > 0, given y = 1 when x = 1.

• Separate: (1/y) dy = 1/(x(x + 1)) dx.
• Partial fractions: 1/(x(x + 1)) = 1/x − 1/(x + 1). Integrate: ln|y| = ln x − ln(x + 1) + c.
• Combine the logs: y = Ax/(x + 1). At x = 1, y = 1: 1 = A/2, so A = 2.
• y = 2x/(x + 1). Writing the constant as ln A instead of c often makes combining logs easier.

Forming differential equations from context

Many questions describe a situation in words and ask you to write the equation. Key phrases translate directly:

• “The rate of change of P is proportional to P”: dP/dt = kP.
• “The rate of decrease of m is proportional to m”: dm/dt = −km, with k > 0.
• “The rate of change of V is inversely proportional to the square root of V”: dV/dt = k/√V.

“Proportional to” introduces a constant k. Use a negative sign when the quantity is decreasing, so that k itself stays positive.

Example 5: population growth

A population grows at a rate proportional to its size. Initially it is 500, and after 2 years it is 800. Find the population after 5 years.

• dP/dt = kP, so P = 500e^kt.
• 800 = 500e^2k, so e^2k = 1.6 and k = ½ ln 1.6 ≈ 0.235.
• P(5) = 500e^5k = 500 × 1.6^2.5 ≈ 1619.

Example 6: Newton's law of cooling

A cup of tea at 90 °C cools in a room at 20 °C. The rate of cooling is proportional to the difference between its temperature θ and the room. After 10 minutes it is 60 °C. Find the time for it to reach 40 °C.

• dθ/dt = −k(θ − 20). Separate: ∫ 1/(θ − 20) dθ = ∫ −k dt, so ln(θ − 20) = −kt + c.
• θ − 20 = Ae^−kt. At t = 0, A = 70. At t = 10: 40 = 70e^−10k, so k = (1/10) ln(7/4) ≈ 0.0560.
• For θ = 40: 20 = 70e^−kt, so t = ln(3.5)/k ≈ 22.4 minutes.

Interpreting and criticising models

Exam questions often ask about the model itself. An exponential growth model predicts that a population grows forever, which is unrealistic because food and space are limited. A cooling model predicts the tea approaches room temperature but never quite reaches it. Comment on whether the model's long-term behaviour makes sense, and suggest a refinement, such as a maximum population. These short answers carry real marks and need context.

Common mistakes

• Forgetting the constant of integration, or adding one on both sides.
• Writing ln y = 3x + c and then y = e^3x + c instead of y = Ae^3x.
• Using the wrong sign for decreasing quantities.
• Not finding the constant from the initial condition before answering the question.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are differential equations in A level Maths?

Yes. First-order equations solved by separating variables, and modelling with them, are in the second year of A level Maths. Second-order equations are in Further Maths.

Are differential equations in AP Calculus?

Yes. Separation of variables and exponential models are in Unit 7 of both AB and BC, and BC adds Euler's method and logistic growth.

Key takeaways

• A differential equation links a quantity to its rate of change.
• Separate variables: y terms with dy, x terms with dx.
• Add one constant of integration, then use the given condition.
• “Proportional to” means multiply by a constant k.
• Comment on the realism of the model in context.

Related guides

• A level Maths integration
• Partial fractions
• A level Maths differentiation