Differential equations by direct integration
Problem 6.584 · easy
Find the general solution of \( \displaystyle y' = x^{2} - 4 \).
- \[ \frac{d}{d x} \left(\frac{x^{3}}{3} - 4 x\right) = x^{2} - 4 \]Integrate both sides; every antiderivative differs by a constant C₁.✓ Proved
Answer \( y = C_{1} + \frac{x^{3}}{3} - 4 x \)
✓ 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 |
| 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 includes the arbitrary constant. The reasoning is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the antiderivative and includes the arbitrary constant. The reasoning is sound.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the antiderivative and includes the arbitrary constant of integration.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the antiderivative and includes the arbitrary constant. The reasoning is sound.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.