- 2071 Magh · 4 marks
Euler: y' = x + y
Short answer
Apply Euler's formula y_{n+1}=y_n+h(x_n+y_n) with h=0.1 from y(0)=0. Two steps give y(0.1)=0 and y(0.2)=0.0100. Thus y(0.2) is approximately 0.0100, underestimating the exact value 0.0214.
Using Euler's method find y(0.2) from following equation y' = x + y, y(0) = 0, take h = 0.1.
Euler's formula: yₙ₊₁ = yₙ + h·f(xₙ, yₙ), with f(x, y) = x + y, x₀ = 0, y₀ = 0, h = 0.1.
Step 1: y(0.1) = y₀ + h(x₀ + y₀) = 0 + 0.1(0 + 0) = 0.0000
Step 2: y(0.2) = y₁ + h(x₁ + y₁) = 0 + 0.1(0.1 + 0) = 0.0100
| n | xₙ | yₙ | f(xₙ,yₙ) | yₙ₊₁ = yₙ + 0.1·f |
|---|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 1 | 0.1000 | 0.0000 | 0.1000 | 0.0100 |
Answer: y(0.2) ≈ 0.0100
The exact solution y = eˣ − x − 1 gives y(0.2) = 0.0214; Euler's method (first order) underestimates here because h is large.
✓ Answer checked by Claude Opus 5.5 (high effort)
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