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Differential equations by direct integration

Problem 6.741 · easy

Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = 8 \).

\[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]
Integrate: y = this + C.✓ Proved
\[ 8 - \left. - x^{2} + \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 8 \]
y(0) = 8 fixes C.✓ Proved
\[ \left. - x^{2} + \sin{\left(x \right)} + 8 \right|_{\substack{ x=0 }} = 8 \]
The initial condition holds.✓ Proved

Answer \( y = - x^{2} + \sin{\left(x \right)} + 8 \)

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: fail (error) — The solution incorrectly derives the constant C. Step 2 uses 8 - Subs(-x**2 + sin(x), x, 0) = 8, which merely shows that the particular antiderivative satisfies the initial condition, but it does not actually determine C. The correct reasoning is C = 8 - (-0^2 + sin 0) = 8. The final answer is correct, but the justification for C is erroneous.
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The steps are logically sound and algebraically verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The steps are logically sound and algebraically verified.
gpt-oss:20b: fail (error) 2026-10-26 — The solution incorrectly derives the constant C. Step 2 uses 8 - Subs(-x**2 + sin(x), x, 0) = 8, which merely shows that the particular antiderivative satisfies the initial condition, but it does not actually determine C. The correct reasoning is C = 8 - (-0^2 + sin 0) = 8. The final answer is correct, but the justification for C is erroneous.
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the antiderivative and verifies the initial condition to determine the constant of integration.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the antiderivative, determines the constant of integration using the initial condition, and verifies the result. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-10-25

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 2026-10-26 with SymPy 1.14.0.