Initial-value problems needing partial fractions
Problem 6.538 · hard
Solve \( \displaystyle \left(t^{2} - 4 t + 3\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(4) = 0 \) (for \( \displaystyle t > 3 \)).
- Separate: dx = dt / (t**2 - 4*t + 3).
- \[ \frac{1}{t^{2} - 4 t + 3} = - \frac{1}{2 t - 2} + \frac{1}{2 t - 6} \]Partial fractions.✓ Proved
- \[ \frac{d}{d t} \frac{\ln{\left(\frac{t - 3}{t - 1} \right)}}{2} = \frac{1}{t^{2} - 4 t + 3} \]An antiderivative, checked by differentiating.✓ Proved
- \[ - \left. \frac{\ln{\left(\frac{t - 3}{t - 1} \right)}}{2} \right|_{\substack{ t=4 }} + 0 = \frac{\ln{\left(3 \right)}}{2} \]x(4) = 0 fixes the constant.✓ Proved
- \[ \left. \frac{\ln{\left(\frac{3 \left(t - 3\right)}{t - 1} \right)}}{2} \right|_{\substack{ t=4 }} = 0 \]The initial condition holds.✓ Proved
Answer \( x(t) = \frac{\ln{\left(\frac{3 \left(t - 3\right)}{t - 1} \right)}}{2} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution ignores the domain constraint t > 3, where t-3 > 0 and t-1 > 0, but fails to handle the absolute values in the logarithm correctly durin
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution ignores the domain constraint t > 3, where t-3 > 0 and t-1 > 0, but fails to handle the absolute values in the logarithm correctly duringpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution ignores the domain constraint t > 3, where (t-3)/(t-1) is positive, leading to an unnecessary absolute value or sign error in the logarithm argument if not handled carefully, but more critically, step 4 implies the constant C is -log(3)/2, yet step 5 shows the final form log(3(t-3)/(t-1))/2. Let's check: x(t) = 1/2 ln((t-3)/(t-1)) + C. x(4) = 1/2 ln(3/3) + C = 1/2 ln(1) + C = C. Since x(4)=0, C=0. The stated answer is 1/2 ln(3(t-3)/(t-1)) = 1/2 ln(3) + 1/2 ln((t-3)/(t-1)). This equals C + 1/2 ln((t-3)/(t-1)) only if C = 1/2 ln(3). But we found C=0. So the stated answer is incorrect. Step 4 calculates -Subs(..., 4) = -1/2 ln(1) = 0, so C=0. Step 5 checks a different function. The solution is inconsistent.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to account for the domain t > 3, where t - 3 > 0 but t - 1 > 0, so the argument of the logarithm (t-3)/(t-1) is positive. Hgpt-oss:20b: pass 2026-10-14
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.