Differential Equations

Revise Differential Equations for Pure Mathematics 3 P3 (A2 Level) — revision notes and instant AI marking.

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Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 3

Differential Equations

The big idea: A differential equation describes how something changes. Separate the variables, integrate both sides, use the initial condition to fix the constant, and then read what the solution says about the situation.

Summary — What This Chapter Covers

Translate a statement about a rate of change into dy/dx (or dy/dt), introducing a constant of proportionality k where needed.
Separable equations: dy/dx = f(x)g(y) ⇒ ∫1/g(y) dy = ∫f(x) dx.
Add one constant of integration, then use the given condition to find it.
Any integration technique from Chapter 5 may be needed for either side.
Interpret: the long-term behaviour, the time to reach a value, whether a quantity increases or decreases.

1. Forming Differential Equations

Modelling

From words to a rate equation

“The rate of change of P” is dP/dt. “Proportional to” means “equals a constant k times”. “Decreasing” means the derivative is negative, so write −k with k > 0. A given rate at a given value lets you find k.

Templates Rate of increase of N is proportional to N: dN/dt = kN Decreasing at a rate proportional to √h: dh/dt = −k√h
Temperature falls at a rate proportional to the excess over 20: dθ/dt = −k(θ − 20)

Worked Example

A population P increases at a rate proportional to P. When P = 100 the population is increasing at 5 per year. Write a differential equation for P.

Practice Question

Water leaks from a tank so that the depth h cm decreases at a rate proportional to the square root of h. Write down a differential equation for h.

2. Separating Variables and Particular Solutions

Solving

Separate, integrate, find c, interpret

Move every y term (with dy) to one side and every x term (with dx) to the other, then integrate both sides with one constant c. Substitute the initial condition to find c. Rearranging to y = … is often required: if ln y = g(x) + c, then y = Ae^g(x).

Method dy/dx = f(x)g(y) ⇒ ∫ dy/g(y) = ∫ f(x) dx + c

Worked Example 1

Solve dy/dx = 6x²y, given y = 2 when x = 0. Express y in terms of x.

Worked Example 2

Practice Question

Solve (x² + 4) dy/dx = y, given y = 1 when x = 0.

Memorise

Proportional

“∝” becomes “= k ×”; decreasing gives −k

Separate

∫dy/g(y) = ∫f(x) dx + c

Constant

Common Mistakes

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Differential Equations