Euler's method
Problem 6.217 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{10} \) to approximate \( \displaystyle y(\frac{3}{10}) \) for \( \displaystyle y' = - t^{2} + y \), \( \displaystyle y(0) = 3 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{10} \cdot 3 + 3 = \frac{33}{10} \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope 3.✓ Proved
- \[ 1 \cdot \frac{1}{10} \cdot 329 \cdot \frac{1}{100} + \frac{33}{10} = \frac{3629}{1000} \]y2 = y1 + h·f(t1, y1) with t1 = 1/10, slope 329/100.✓ Proved
- \[ 1 \cdot \frac{1}{10} \cdot 3589 \cdot \frac{1}{1000} + \frac{3629}{1000} = \frac{39879}{10000} \]y3 = y2 + h·f(t2, y2) with t2 = 1/5, slope 3589/1000.✓ Proved
Answer \( y(\frac{3}{10}) \approx \frac{39879}{10000} = 3.9879 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the steps rerun in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies Euler's method with the specified step size and initial conditions, and the arithmetic in the proven steps is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies Euler's method with the specified step size and initial conditions, and the arithmetic in the proven steps is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/eulers_method, checked 2026-10-04 with SymPy 1.14.0.