Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.541

Initial-value problems needing partial fractions

Problem 6.541 · hard

Solve \( \displaystyle \left(t^{2} - 4 t\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(5) = 0 \) (for \( \displaystyle t > 4 \)).

Separate: dx = dt / (t**2 - 4*t).
\[ \frac{1}{t^{2} - 4 t} = \frac{1}{4 t - 16} - \frac{1}{4 t} \]
Partial fractions.✓ Proved
\[ \frac{d}{d t} \frac{\ln{\left(\frac{t - 4}{t} \right)}}{4} = \frac{1}{t^{2} - 4 t} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - \left. \frac{\ln{\left(\frac{t - 4}{t} \right)}}{4} \right|_{\substack{ t=5 }} + 0 = \frac{\ln{\left(5 \right)}}{4} \]
x(5) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\ln{\left(5 - \frac{20}{t} \right)}}{4} \right|_{\substack{ t=5 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\ln{\left(5 - \frac{20}{t} \right)}}{4} \)

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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithmic integral, which is necessary because the argument (t-4)/t is negative for 4 < t < 5. Consequently, the derived formula x(t) = ln(5 - 20/t)/4 is undefined for 4 < t < 5, failing to solve the problem on the entire specified domain t > 4.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution fails to account for the absolute value in the logarithmic integral, which is necessary because the argument (t-4)/t is negative for 4 < t < 5. Consequently, the derived formula x(t) = ln(5 - 20/t)/4 is undefined for 4 < t < 5, failing to solve the problem on the entire specified domain t > 4.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to account for the absolute value in the integral of 1/t, which is critical for the domain t > 4. Specifically, for t > 4, (t-4)/
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution contains a sign error in the partial fraction decomposition. The correct decomposition of 1/(t^2 - 4t) is 1/(4t) - 1/(4(t-4)), which integrates to (1/4)ln|t/(t-4)|. The solution uses the negative of this, leading to an incorrect final answer.
gpt-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.