Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.691

Initial-value problems needing partial fractions

Problem 6.691 · hard

Solve \( \displaystyle \left(t^{2} - 2 t - 15\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(7) = 0 \) (for \( \displaystyle t > 5 \)).

Separate: dx = dt / (t**2 - 2*t - 15).
\[ \frac{1}{t^{2} - 2 t - 15} = - \frac{1}{8 t + 24} + \frac{1}{8 t - 40} \]
Partial fractions.✓ Proved
\[ \frac{d}{d t} \frac{\ln{\left(\frac{t - 5}{t + 3} \right)}}{8} = \frac{1}{t^{2} - 2 t - 15} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - \left. \frac{\ln{\left(\frac{t - 5}{t + 3} \right)}}{8} \right|_{\substack{ t=7 }} + 0 = \frac{\ln{\left(5 \right)}}{8} \]
x(7) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\ln{\left(\frac{5 \left(t - 5\right)}{t + 3} \right)}}{8} \right|_{\substack{ t=7 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\ln{\left(\frac{5 \left(t - 5\right)}{t + 3} \right)}}{8} \)

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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution ignores the domain constraint t > 5, which requires the argument of the logarithm to be positive. It should use ln((t-5)/(t+3)) or ln((5

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution ignores the domain constraint t > 5, which requires the argument of the logarithm to be positive. It should use ln((t-5)/(t+3)) or ln((5
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution ignores the domain constraint t > 5, which implies t - 5 > 0. Consequently, it fails to use absolute values in the logarithm (ln|t-5|), leading to an incorrect constant of integration. The correct solution should be x(t) = (1/8)ln((t-5)/(t+3)), whereas the provided answer includes an extraneous factor of 5 inside the logarithm.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-17 — 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: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.