Euler's method
Problem 6.335 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{5} \) to approximate \( \displaystyle y(\frac{2}{5}) \) for \( \displaystyle y' = 1 - y \), \( \displaystyle y(0) = 1 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{5} \cdot 0 + 1 = 1 \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope 0.✓ Proved
- \[ 1 \cdot \frac{1}{5} \cdot 0 + 1 = 1 \]y2 = y1 + h·f(t1, y1) with t1 = 1/5, slope 0.✓ Proved
Answer \( y(\frac{2}{5}) \approx 1 = 1 \)
✓ Nihil obstat Lines: 2 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 |
| 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. Since y(0)=1 and y'=1-y, the slope is 0 at the first step, keeping y constant at 1 for subsequent steps. The logic and calculations are sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies Euler's method. Since y(0)=1 and y'=1-y, the slope is 0 at the first step, keeping y constant at 1 for subsequent steps. The logic and calculations are sound.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies Euler's method. Since y(0)=1 and y'=1-y, the slope is 0 at the first step, keeping y at 1. The second step also has slope 0, so y remains 1. The approximation y(2/5) ≈ 1 is correct.
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 2026-10-09 with SymPy 1.14.0.