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Differential equations by direct integration

Problem 6.573 · medium

Find the general solution of \( \displaystyle y'' = \frac{1}{x} \).

\[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]
Integrate once: y′ = log(x) + C₁.✓ Proved
\[ \frac{d}{d x} \left(x \ln{\left(x \right)} - x\right) = \ln{\left(x \right)} \]
Integrate again: y = x*log(x) - x + C₁x + C₂.✓ Proved

Answer \( y = C_{1} x + C_{2} + x \ln{\left(x \right)} - x \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
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: pass — The solution correctly integrates the differential equation twice, applying the power rule and logarithmic integration rules accurately. The final general solution matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly integrates the differential equation twice, applying the power rule and logarithmic integration rules accurately. The final general solution matches the stated answer.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the fundamental theorem of calculus twice to find the general solution. The integration steps and the final form are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly integrates y'' = 1/x twice, handling the integration constants properly to arrive at the general solution.
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/de_direct_integration, checked 2026-10-18 with SymPy 1.14.0.