Home›Calculus 2›Initial-value problems needing partial fractions›Problem 6.461

Initial-value problems needing partial fractions

Problem 6.461 · medium

Solve \( \displaystyle \left(t^{2} + 36\right)\dfrac{dx}{dt} = 1 \) with \( \displaystyle x(0) = 0 \).

Separate: dx = dt / (t**2 + 36).
Reviewed
1/(t² + 36) integrates to (1/6) arctan(t/6).
Reviewed
\[ \frac{d}{d t} \frac{\operatorname{atan}{\left(\frac{t}{6} \right)}}{6} = \frac{1}{t^{2} + 36} \]
An antiderivative, checked by differentiating.✓ Proved
\[ - \left. \frac{\operatorname{atan}{\left(\frac{t}{6} \right)}}{6} \right|_{\substack{ t=0 }} + 0 = 0 \]
x(0) = 0 fixes the constant.✓ Proved
\[ \left. \frac{\operatorname{atan}{\left(\frac{t}{6} \right)}}{6} \right|_{\substack{ t=0 }} = 0 \]
The initial condition holds.✓ Proved

Answer \( x(t) = \frac{\operatorname{atan}{\left(\frac{t}{6} \right)}}{6} \)

Lines: 3 proved, 2 reviewed. 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	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 form for arctan, and verifies the initial condition. The logic is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly separates variables, identifies the standard integral form for arctan, and verifies the initial condition. The logic is sound and the steps are clear.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly separates variables, identifies the standard integral form for arctan, and verifies the initial condition. The logic is sound and the steps are clearly justified.
gpt-oss:20b: fail (error) 2026-10-14 — The solution omits the constant of integration when integrating 1/(t²+36). The antiderivative should be (1/6)arctan(t/6)+C, and then C is determined by x(0)=0. Without including C the reasoning is incomplete and could mislead a student.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly separates variables, identifies the standard integral form for arctan, and applies the initial condition to determine the constant of integration. The steps are logically sound and algebraically correct.
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.