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Differential equations by direct integration

Problem 6.526 · easy

Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 6 \).

\[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]
Integrate: y = this + C.✓ Proved
\[ 6 - \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 6 \]
y(1) = 6 fixes C.✓ Proved
\[ \left. \ln{\left(x \right)} + 6 \right|_{\substack{ x=1 }} = 6 \]
The initial condition holds.✓ Proved

Answer \( y = \ln{\left(x \right)} + 6 \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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	finite differences of the answer match f, and the initial value holds

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to explicitly state the general solution y = ln(x) + C before applying the initial condition. It jumps from the derivative check

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to explicitly state the general solution y = ln(x) + C before applying the initial condition. It jumps from the derivative check
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The logic is sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly solve for the constant of integration C. While the final answer is correct, the steps shown are merely verification checks rather than a derivation, omitting the crucial step where C is determined to be 5.
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/de_direct_integration, checked 2026-10-18 with SymPy 1.14.0.