Key Idea: Euler's method estimates a solution of dy/dx = f(x, y) in small steps. You use it when the exact answer is hard, then check how far the estimate is from the truth.
| Idea | What to do |
|---|---|
| Recurrence | yₙ₊₁ = yₙ + h·f(xₙ, yₙ) and xₙ₊₁ = xₙ + h |
| Steps | (target x − x₀) ÷ h; count first |
| Accuracy | Smaller h: more steps, closer result |
| Error | |approx − exact| ÷ exact × 100% |
Insects: dP/dt = 0.15P, P = 200 at t = 0, h = 0.2
- Count — 0.6 ÷ 0.2 = 3 steps.
- Step 1 — P₁ = 200 + 0.2(30) = 206.
- Step 2 — P₂ = 206 + 0.2(30.9) = 212.18.
- Step 3 — P₃ = 218.545. Exact: 218.835, so the error is about 0.13%.
Carry full accuracy in every step and round only the final answer.
From x = 1 to x = 2 with h = 0.25. How many steps? 4.
How do you make an Euler estimate more accurate? Use a smaller h.
dy/dx = x + y, start (0, 1), h = 0.1. Find y₁. 1 + 0.1(0 + 1) = 1.1.
Was the insect estimate above or below the exact value? Below: 218.545 < 218.835.