Euler's method
Problem 6.212 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{2} \) to approximate \( \displaystyle y(2) \) for \( \displaystyle y' = t y \), \( \displaystyle y(0) = 1 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{2} \cdot 0 + 1 = 1 \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope 0.✓ Proved
- \[ 1 \cdot \frac{1}{2} \cdot 1 \cdot \frac{1}{2} + 1 = \frac{5}{4} \]y2 = y1 + h·f(t1, y1) with t1 = 1/2, slope 1/2.✓ Proved
- \[ 1 \cdot \frac{1}{2} \cdot 5 \cdot \frac{1}{4} + \frac{5}{4} = \frac{15}{8} \]y3 = y2 + h·f(t2, y2) with t2 = 1, slope 5/4.✓ Proved
- \[ 1 \cdot \frac{1}{2} \cdot 45 \cdot \frac{1}{16} + \frac{15}{8} = \frac{105}{32} \]y4 = y3 + h·f(t3, y3) with t3 = 3/2, slope 45/16.✓ Proved
Answer \( y(2) \approx \frac{105}{32} = 3.28125 \)
✓ Nihil obstat Lines: 4 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 |
| 5 | ✓ 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. The arithmetic in each step is verified as correct, and the final approximation matches the stated answer.
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. The arithmetic in each step is verified as correct, and the final approximation matches the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies Euler's method with the specified step size and initial conditions. The arithmetic in each step is correct, and the final approximation matches the stated answer.gpt-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.