Euler's method
Problem 6.375 · medium
Use Euler's method with step size \( \displaystyle h = \frac{1}{5} \) to approximate \( \displaystyle y(\frac{7}{5}) \) for \( \displaystyle y' = t y \), \( \displaystyle y(1) = 1 \).
- Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.Reviewed
- \[ 1 \cdot \frac{1}{5} \cdot 1 + 1 = \frac{6}{5} \]y1 = y0 + h·f(t0, y0) with t0 = 1, slope 1.✓ Proved
- \[ 1 \cdot \frac{1}{5} \cdot 36 \cdot \frac{1}{25} + \frac{6}{5} = \frac{186}{125} \]y2 = y1 + h·f(t1, y1) with t1 = 6/5, slope 36/25.✓ Proved
Answer \( y(\frac{7}{5}) \approx \frac{186}{125} = 1.488 \)
Lines: 2 proved, 1 reviewed. 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 | 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 with the specified step size and initial conditions. The intermediate calculations for y1 and y2 are accurate, and the final approximation matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies Euler's method with the specified step size and initial conditions. The intermediate calculations for y1 and y2 are accurate, and the final approximation matches the stated answer.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution performs only two steps, reaching t = 1.4 (7/5), but the calculation for the second step is incorrect. The slope at step 1 should be f(t1, y1) = (6/5)*(6/5) = 36/25, which is correct, but the update y2 = y1 + h*f(t1, y1) = 6/5 + (1/5)*(36/25) = 30/25 + 36/125 = 150/125 + 36/125 = 186/125 is actually correct arithmetic. Wait, let me re-read carefully. Step 0: t0=1, y0=1. f(1,1)=1. y1 = 1 + (1/5)*1 = 6/5. t1 = 1 + 1/5 = 6/5. Step 1: t1=6/5, y1=6/5. f(6/5, 6/5) = (6/5)*(6/5) = 36/25. y2 = 6/5 + (1/5)*(36/25) = 6/5 + 36/125 = 150/125 + 36/125 = 186/125. t2 = 6/5 + 1/5 = 7/5. The arithmetic is correct. The number of steps is correct (2 steps to get from 1 to 1.4 with h=0.2). The result 186/125 is correct. Why did I think it was wrong? Let me double check the problem statement. y' = ty, y(1)=1. Approximate y(7/5). h=1/5. t0=1, y0=1. t1=1.2, y1=1.2. t2=1.4, y2=1.2 + 0.2*(1.2*1.2) = 1.2 + 0.2*1.44 = 1.2 + 0.288 = 1.488. 186/125 = 1.488. The solution is actually correct. The sentences are minimal but correct. The equations are marked 'unchecked' in the prompt description but the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The prompt asks me to review the sentences. Sentence 1: 'Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.' This is the correct definition of Euler's method. The equations are labeled [equation, unchecked] in the text provided by the user, but the instructions say 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. The user provided text has 'unchecked' in the brackets, which might be a typo in the user's prompt or indicating that I should check them? No, the instructions say 'do not re-check the algebra of a line marked proved'. Here they are marked 'unchecked'. This is ambiguous. However, usually in these tasks, if it's labeled equation, I assume the algebra is correct unless it's obviously wrong. Here the algebra is correct. Is there any error in the setup? - Right formula: Yes. - Right quantities: Yes. - Right bounds: Yes (1 to 1.4). - Right direction: Yes. The solution is sound. I will mark it as pass.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.