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Initial-value problems needing partial fractions

Problem 6.456 · medium

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

Separate: dx = dt / (t**2 + 4).
Reviewed
1/(t² + 4) integrates to (1/2) arctan(t/2).
Reviewed
\[ \frac{d}{d t} \frac{\operatorname{atan}{\left(\frac{t}{2} \right)}}{2} = \frac{1}{t^{2} + 4} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - \left. \frac{\operatorname{atan}{\left(\frac{t}{2} \right)}}{2} \right|_{\substack{ t=0 }} + 0 = 0 \]
x(0) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\operatorname{atan}{\left(\frac{t}{2} \right)}}{2} \right|_{\substack{ t=0 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\operatorname{atan}{\left(\frac{t}{2} \right)}}{2} \)

✓ Nihil obstat Lines: 3 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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 — The solution correctly applies separation of variables and the standard arctangent integration formula. The initial condition is handled correctly, and the final answer matches the stated solution.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies separation of variables and the standard arctangent integration formula. The initial condition is handled correctly, and the final answer matches the stated solution.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14
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.