Differential equations by direct integration
Problem 6.432 · easy
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = 10 \).
- \[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]Integrate: y = this + C.✓ Proved
- \[ 10 - \left. - x^{2} + \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 10 \]y(0) = 10 fixes C.✓ Proved
- \[ \left. - x^{2} + \sin{\left(x \right)} + 10 \right|_{\substack{ x=0 }} = 10 \]The initial condition holds.✓ Proved
Answer \( y = - x^{2} + \sin{\left(x \right)} + 10 \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The logic is sound and the steps are verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The logic is sound and the steps are verified.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative and verifies the initial condition to determine the constant of integration.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The algebraic checks confirm the result.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 2026-10-17 with SymPy 1.14.0.