Euler's method
Problem 6.219 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{5} \) to approximate \( \displaystyle y(\frac{4}{5}) \) for \( \displaystyle y' = t - y \), \( \displaystyle y(0) = 2 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{5} \left(-2\right) + 2 = \frac{8}{5} \]y1 = y0 + h·f(t0, y0) with t0 = 0, slope -2.✓ Proved
- \[ 1 \cdot \frac{1}{5} \left(-7\right) \frac{1}{5} + \frac{8}{5} = \frac{33}{25} \]y2 = y1 + h·f(t1, y1) with t1 = 1/5, slope -7/5.✓ Proved
- \[ 1 \cdot \frac{1}{5} \left(-23\right) \frac{1}{25} + \frac{33}{25} = \frac{142}{125} \]y3 = y2 + h·f(t2, y2) with t2 = 2/5, slope -23/25.✓ Proved
- \[ 1 \cdot \frac{1}{5} \left(-67\right) \frac{1}{125} + \frac{142}{125} = \frac{643}{625} \]y4 = y3 + h·f(t3, y3) with t3 = 3/5, slope -67/125.✓ Proved
Answer \( y(\frac{4}{5}) \approx \frac{643}{625} = 1.0288 \)
✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-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.