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Differential equations by direct integration

Problem 6.534 · easy

Solve \( \displaystyle y' = 6 x \) with \( \displaystyle y(0) = 9 \).

\[ \frac{d}{d x} 3 x^{2} = 6 x \]
Integrate: y = this + C.✓ Proved
\[ 9 - \left. 3 x^{2} \right|_{\substack{ x=0 }} = 9 \]
y(0) = 9 fixes C.✓ Proved
\[ \left. 3 x^{2} + 9 \right|_{\substack{ x=0 }} = 9 \]
The initial condition holds.✓ Proved

Answer \( y = 3 x^{2} + 9 \)

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: pass — The solution correctly identifies the antiderivative, applies the initial condition to determine the constant, and verifies the result. The logic is sound and complete.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the antiderivative, applies the initial condition to determine the constant, and verifies the result. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the general antiderivative, determines the constant of integration using the initial condition, and verifies the result. The logic is sound and complete.
gpt-oss:20b: fail (misleading) 2026-10-15 — Step 2 incorrectly derives the constant C; it shows 9-0=9 instead of solving 3*0^2+C=9 to find C=9. This could mislead a student about how to determine the integration constant.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative and verifies the initial condition. The comments accurately describe the logical steps taken.
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.