First-Order Differential Equations

Revise First-Order Differential Equations for Further Pure Mathematics 2 WFM02 (A2 Level) — revision notes and instant AI marking.

📖 Revision notes · preview

Edexcel IAL Further Pure Mathematics 2 (WFM02)

First-Order Differential Equations

The big idea: Every first-order method aims for the same thing: an exact derivative on one side, so you can integrate both sides directly.

Summary — What This Chapter Covers

Separable: dy/dx = f(x)g(y) ⇒ ∫ 1/g(y) dy = ∫ f(x) dx. The constant gives a family of solution curves.
Linear: dy/dx + P(x)y = Q(x). Multiply by the integrating factor I = e^{∫P dx}.
Then d/dx(Iy) = IQ, so Iy = ∫ IQ dx.
A given substitution turns an unfamiliar equation into a separable or linear one.
Use initial conditions to find the particular solution.

1. Separable Variables and Families of Solution Curves

Separable

Separate, integrate, add one constant

Collect y terms with dy and x terms with dx, integrate both sides, and add a single constant. Each value of the constant gives one member of the family of solution curves.

Worked Example

Find the general solution of dy/dx = xy and describe the family of curves.

2. The Integrating Factor Method

Integrating factor

dy/dx + Py = Q

Make sure the coefficient of dy/dx is 1 first. Then the integrating factor makes the left side the derivative of a product.

Method I = e^{∫P dx} ⇒ d/dx(Iy) = IQ ⇒ Iy = ∫ IQ dx + c

Worked Example

Solve dy/dx + 2y/x = x, x > 0.

Practice Question 1

Solve dy/dx + y tan x = sec x, 0 < x < π/2.

3. Substitutions for First-Order Equations

Substitution

Following the given substitution

The question always gives the substitution. Differentiate it to replace dy/dx, simplify to a separable or linear equation, solve, then substitute back to give y in terms of x.

Worked Example

Use z = 1/y to solve dy/dx + y/x = xy².

Memorise

Integrating factor

e^{∫P dx} — coefficient of dy/dx must be 1 first

Useful IFs

🔓 Read the full First-Order Differential Equations note → You're seeing the preview · sign in to read it all

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

📄 Practise First-Order Differential Equations with Further Pure Mathematics 2 WFM02 past papers Every paper with its mark scheme — answer online, marked instantly. Open →

Read the full First-Order Differential Equations notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →

First-Order Differential Equations