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Initial-value problems needing partial fractions

Problem 6.608 · hard

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

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

Answer \( x(t) = \ln{\left(\frac{2 t}{t + 1} \right)} \)

✓ 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 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly separates variables, identifies the antiderivative, and determines the constant of integration to satisfy the initial condition. The final expression matches the stated answer.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly separates variables, applies partial fraction decomposition, and determines the constant of integration to satisfy the initial condition. The final result matches the stated answer.
gpt-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.