Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.464

Initial-value problems needing partial fractions

Problem 6.464 · hard

Solve \( \displaystyle \left(t^{2} - 4 t\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(7) = 0 \) (for \( \displaystyle t > 4 \)).

Separate: dx = dt / (t**2 - 4*t).
Reviewed
\[ \frac{1}{t^{2} - 4 t} = \frac{1}{4 t - 16} - \frac{1}{4 t} \]
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
\[ - \left. \frac{\ln{\left(\frac{t - 4}{t} \right)}}{4} \right|_{\substack{ t=7 }} + 0 = - \frac{\ln{\left(3 \right)}}{4} + \frac{\ln{\left(7 \right)}}{4} \]
x(7) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\ln{\left(\frac{7 \left(t - 4\right)}{3 t} \right)}}{4} \right|_{\substack{ t=7 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\ln{\left(\frac{7 \left(t - 4\right)}{3 t} \right)}}{4} \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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
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: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies separation of variables, partial fraction decomposition, and integration. The handling of the initial condition and the domain t > 4 is correct, ensuring the argument of the logarithm is positive.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.