Key Idea: A differential equation links y and dy/dx. This topic solves them by separating, by an integrating factor, or by y = vx, and estimates them with Euler's method. They appear in Paper 1, Paper 2 and Paper 3.
Separation of variables
- Check — dy/dx must be (x-part)·(y-part). A sum like x + y is not.
- Split — y's with dy on the left, x's with dx on the right.
- Integrate — Both sides, with one + C.
- Find C — Use the initial condition, then rearrange for y.
| Equation looks like | Method |
|---|---|
| dy/dx + P(x)y = Q(x) | Multiply by I = e∫P ᵈˣ; then (Iy)′ = IQ |
| dy/dx uses only y/x | y = vx, dy/dx = v + x dv/dx, then separate |
| Already separable | Separate straight away |
Euler's method
- yₙ+₁ = yₙ + h·f(xₙ, yₙ) and xₙ+₁ = xₙ + h.
- Smaller h: better estimate, more steps.
- Concave up: underestimate. Concave down: overestimate.
Always check your answer with the initial point.
Solve dy/dx = x/y with y = 4 when x = 0. y = √(x² + 16).
Integrating factor for dy/dx + (cot x)y = 1? sin x.
dy/dx = x·y², y = 1 at x = 1, h = 0.25. Estimate y(1.25). 1.25.
Why is Euler only an estimate? It uses the gradient at the start of each step.