Euler's method
Problem 6.254 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{2} \) to approximate \( \displaystyle y(3) \) for \( \displaystyle y' = 1 - y \), \( \displaystyle y(1) = 2 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{2} \left(-1\right) + 2 = \frac{3}{2} \]y1 = y0 + h·f(t0, y0) with t0 = 1, slope -1.✓ Proved
- \[ 1 \cdot \frac{1}{2} \left(-1\right) \frac{1}{2} + \frac{3}{2} = \frac{5}{4} \]y2 = y1 + h·f(t1, y1) with t1 = 3/2, slope -1/2.✓ Proved
- \[ 1 \cdot \frac{1}{2} \left(-1\right) \frac{1}{4} + \frac{5}{4} = \frac{9}{8} \]y3 = y2 + h·f(t2, y2) with t2 = 2, slope -1/4.✓ Proved
- \[ 1 \cdot \frac{1}{2} \left(-1\right) \frac{1}{8} + \frac{9}{8} = \frac{17}{16} \]y4 = y3 + h·f(t3, y3) with t3 = 5/2, slope -1/8.✓ Proved
Answer \( y(3) \approx \frac{17}{16} = 1.0625 \)
✓ 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, arriving at the correct approximation for y(3).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies Euler's method with the specified step size and initial conditions, arriving at the correct approximation for y(3).gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.