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Differential equations by direct integration

Problem 6.527 · easy

Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(1) = 5 \).

\[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]
Integrate: y = this + C.✓ Proved
\[ 5 - \left. - x^{2} + \sin{\left(x \right)} \right|_{\substack{ x=1 }} = 6 - \sin{\left(1 \right)} \]
y(1) = 5 fixes C.✓ Proved
\[ \left. - x^{2} + \sin{\left(x \right)} - \sin{\left(1 \right)} + 6 \right|_{\substack{ x=1 }} = 5 \]
The initial condition holds.✓ Proved

Answer \( y = - x^{2} + \sin{\left(x \right)} - \sin{\left(1 \right)} + 6 \)

✓ 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
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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and verifies the result. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and verifies the result. The logic is sound and complete.
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.