Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.606

Initial-value problems needing partial fractions

Problem 6.606 · medium

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

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

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

✓ 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 separates variables, identifies the standard integral, and applies the initial condition. The logic is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly separates variables, identifies the standard integral, and applies the initial condition. The logic is sound and the steps are clear.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly separates variables, identifies the standard integral, and applies the initial condition to determine the constant of integration.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly separates variables, identifies the standard integral of 1/(t^2+1) as arctan(t), and correctly applies the initial condition x(0)=0 to determine the constant of integration is 0.
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.