Initial-value problems needing partial fractions
Problem 6.613 · hard
Solve \( \displaystyle \left(t^{2} - 6 t + 8\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(6) = 0 \) (for \( \displaystyle t > 4 \)).
- Separate: dx = dt / (t**2 - 6*t + 8).
- \[ \frac{1}{t^{2} - 6 t + 8} = - \frac{1}{2 t - 4} + \frac{1}{2 t - 8} \]Partial fractions.✓ Proved
- \[ \frac{d}{d t} \frac{\ln{\left(\frac{t - 4}{t - 2} \right)}}{2} = \frac{1}{t^{2} - 6 t + 8} \]An antiderivative, checked by differentiating.✓ Proved
- \[ - \left. \frac{\ln{\left(\frac{t - 4}{t - 2} \right)}}{2} \right|_{\substack{ t=6 }} + 0 = \frac{\ln{\left(2 \right)}}{2} \]x(6) = 0 fixes the constant.✓ Proved
- \[ \left. \frac{\ln{\left(\frac{2 \left(t - 4\right)}{t - 2} \right)}}{2} \right|_{\substack{ t=6 }} = 0 \]The initial condition holds.✓ Proved
Answer \( x(t) = \frac{\ln{\left(\frac{2 \left(t - 4\right)}{t - 2} \right)}}{2} \)
Lines: 4 proved, 1 not checked. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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 | mpmath's ODE solver from t₀ to t₀ + 1 |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution’s constant‑term reasoning is incorrect: it claims that \(-\int_{6}^{t}\!1/(s^2-6s+8)\,ds= \ln 2/2\), which is the opposite sign of the aqwen3.6:27b-mlx: pass — The solution correctly separates variables, applies partial fraction decomposition, and determines the constant of integration to satisfy the initial condition. The final expression matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly separates variables, applies partial fraction decomposition, and determines the constant of integration to satisfy the initial condition. The final expression matches the stated answer.gpt-oss:20b: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The solution’s constant‑term reasoning is incorrect: it claims that \(-\int_{6}^{t}\!1/(s^2-6s+8)\,ds= \ln 2/2\), which is the opposite sign of the aqwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithm arising from the integration of 1/(t-2) and 1/(t-4). For t > 4, both (t-2) and (t-4) are positive, so the absolute values are redundant, but the logic in step 4 implies a specific sign choice without justification. More critically, step 4 calculates the constant C by evaluating the antiderivative at t=6 and setting x(6)=0, but it does so incorrectly: it computes -F(6) + 0 = log(2)/2, implying C = log(2)/2. However, the final answer in step 5 uses a form log(2(t-4)/(t-2))/2 which is equivalent to log((t-4)/(t-2))/2 + log(2)/2. While the final answer is correct, the derivation in step 4 is confusingly presented as '-Subs(...) + 0 = log(2)/2' which looks like it's solving for C but the notation is ambiguous. Actually, looking closer: Step 3 gives F(t) = log((t-4)/(t-2))/2. Step 4 says -F(6) + 0 = log(2)/2. F(6) = log(4/4)/2 = 0. So -0 + 0 = 0. But it claims the result is log(2)/2. This is a calculation error in the sentence description or the equation itself. F(6) = log((6-4)/(6-2))/2 = log(2/4)/2 = log(1/2)/2 = -log(2)/2. So -F(6) = log(2)/2. The equation says '-Subs(...) + 0 = log(2)/2'. If Subs is F(6), then -F(6) = log(2)/2 is correct. But the final answer is x(t) = F(t) + C. If C = log(2)/2, then x(t) = log((t-4)/(t-2))/2 + log(2)/2 = log(2(t-4)/(t-2))/2. This matches. So the math is actually correct. Why did I think it was wrong? Let's re-read carefully. Step 4: '-Subs(log((t - 4)/(t - 2))/2, t, 6) + 0 = log(2)/2'. This is an equation stating that the constant C is log(2)/2. It is derived from x(6) = F(6) + C = 0 => C = -F(6). F(6) = log(2/4)/2 = -log(2)/2. So C = log(2)/2. The equation holds. Step 5 checks the final answer. The solution is sound. Wait, is there a domain issue? t > 4. t-2 > 2, t-4 > 0. Arguments of log are positive. No absolute value needed. The steps are logically consistent. The 'sentence' in step 1 is just 'Separate...'. Step 2, 3, 4, 5 are equations. The prompt asks to review sentences. Step 1 is a sentence. Is it correct? Yes. Are there other sentences? No. The equations are marked 'proved'. So I should only check if the sentences are correct. Step 1 is correct. The equations are checked by CAS. So it should pass? Let me double check the 'proved' status. If the CAS checked the equations, I don't need to re-check them. But I need to check if the sentences are misleading or wrong. Step 1 is fine. Is there a hidden sentence? No. So it passes.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to account for the domain t > 4, where t - 2 > 0 but t - 4 > 0, so the argument of the logarithm (t-4)/(t-2) is positive. Howevergpt-oss:20b: pass 2026-10-15
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/ivp_partial_fractions, checked 2026-10-18 with SymPy 1.14.0.