Home›Calculus 2›Euler's method›Problem 6.339

Euler's method

Problem 6.339 · medium

Use Euler's method with step size \( \displaystyle h = \frac{1}{4} \) to approximate \( \displaystyle y(\frac{7}{4}) \) 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
\[ -1 + 1 \cdot \frac{1}{4} \cdot 2 = - \frac{1}{2} \]
y1 = y0 + h·f(t0, y0) with t0 = 1, slope 2.✓ Proved
\[ - \frac{1}{2} + 1 \cdot \frac{1}{4} \cdot 7 \cdot \frac{1}{4} = - \frac{1}{16} \]
y2 = y1 + h·f(t1, y1) with t1 = 5/4, slope 7/4.✓ Proved
\[ - \frac{1}{16} + 1 \cdot \frac{1}{4} \cdot 25 \cdot \frac{1}{16} = \frac{21}{64} \]
y3 = y2 + h·f(t2, y2) with t2 = 3/2, slope 25/16.✓ Proved

Answer \( y(\frac{7}{4}) \approx \frac{21}{64} = 0.328125 \)

Lines: 3 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
4	✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies Euler's method with the specified step size and initial conditions. The intermediate calculations for slopes and updates are accurate, leading to the correct final approximation.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies Euler's method with the specified step size and initial conditions. The intermediate calculations for slopes and updates are accurate, leading to the correct final approximation.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution calculates y(7/4) but claims the slope at step 2 is 25/16. The actual slope f(t2, y2) = t2 - y2 = 3/2 - (-1/16) = 25/16 is correct, but the previous step's calculation of y2 is wrong. Let's re-verify: y0=-1, t0=1. f(1,-1)=2. y1 = -1 + 1/4(2) = -1/2. t1=5/4. f(5/4, -1/2) = 5/4 - (-1/2) = 7/4. y2 = -1/2 + 1/4(7/4) = -2/4 + 7/16 = -8/16 + 7/16 = -1/16. t2=3/2. f(3/2, -1/16) = 3/2 - (-1/16) = 24/16 + 1/16 = 25/16. y3 = -1/16 + 1/4(25/16) = -1/16 + 25/64 = -4/64 + 25/64 = 21/64. The calculations are actually correct. Wait, let me re-read the prompt's instruction. 'Each line of the solution is either an EQUATION... or a SENTENCE'. The equations are marked 'unchecked' in the prompt description but the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. However, the lines are marked '[equation, unchecked]'. This is a contradiction in the prompt's meta-instructions vs the specific tags. Usually, 'unchecked' means I should check it. If I check it, the math is correct. Let's look closer. t0=1, y0=-1. h=1/4. Step 1: t1 = 1.25. y1 = -1 + 0.25(1 - (-1)) = -1 + 0.5 = -0.5. Step 2: t2 = 1.5. y2 = -0.5 + 0.25(1.25 - (-0.5)) = -0.5 + 0.25(1.75) = -0.5 + 0.4375 = -0.0625 = -1/16. Step 3: t3 = 1.75 = 7/4. y3 = -0.0625 + 0.25(1.5 - (-0.0625)) = -0.0625 + 0.25(1.5625) = -0.0625 + 0.390625 = 0.328125 = 21/64. The math is correct. Why would it fail? Maybe the sentence? 'Each step follows the tangent line...' is a bit loose but standard. Is there a hidden error? Let's check the slope calculation in line 3. t1=5/4, y1=-1/2. f(t1,y1) = 5/4 - (-1/2) = 5/4 + 2/4 = 7/4. Correct. Line 4: t2=3/2, y2=-1/16. f(t2,y2) = 3/2 - (-1/16) = 24/16 + 1/16 = 25/16. Correct. The result is correct. The setup is correct. The steps are correct. I will pass it.
gpt-oss:20b: pass 2026-10-09

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.