Euler's method
Problem 6.293 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{4} \) to approximate \( \displaystyle y(\frac{1}{2}) \) for \( \displaystyle y' = t + y \), \( \displaystyle y(0) = 0 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.
- \[ 0 + 1 \cdot \frac{1}{4} \cdot 0 = 0 \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope 0.✓ Proved
- \[ 0 + 1 \cdot \frac{1}{4} \cdot 1 \cdot \frac{1}{4} = \frac{1}{16} \]y2 = y1 + h·f(t1, y1) with t1 = 1/4, slope 1/4.✓ Proved
Answer \( y(\frac{1}{2}) \approx \frac{1}{16} = 0.0625 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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 |
| 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: fail (error) — The solution stops after two steps, providing an approximation for y(1/4) instead of the requested y(1/2). A third step is required to reach t = 1/2.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution stops after two steps, providing an approximation for y(1/4) instead of the requested y(1/2). A third step is required to reach t = 1/2.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution stops after two steps, providing an approximation for y(1/4) instead of the requested y(1/2). A third step is required to reach t = 1/2.
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 2026-10-07 with SymPy 1.14.0.