Differential equations by direct integration
Problem 6.582 · easy
Solve \( \displaystyle y' = x^{2} - 2 \) with \( \displaystyle y(0) = 8 \).
- \[ \frac{d}{d x} \left(\frac{x^{3}}{3} - 2 x\right) = x^{2} - 2 \]Integrate: y = this + C.✓ Proved
- \[ 8 - \left. \frac{x^{3}}{3} - 2 x \right|_{\substack{ x=0 }} = 8 \]y(0) = 8 fixes C.✓ Proved
- \[ \left. \frac{x^{3}}{3} - 2 x + 8 \right|_{\substack{ x=0 }} = 8 \]The initial condition holds.✓ Proved
Answer \( y = \frac{x^{3}}{3} - 2 x + 8 \)
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 (misleading) — Step 2 incorrectly states "8 - Subs(x**3/3 - 2*x, x, 0) = 8"; the correct reasoning is that y(0)=C, so C=8. The expression as written is misleading and could confuse a student.qwen3.6:27b-mlx: pass — The solution correctly integrates the differential equation and applies the initial condition to determine the constant of integration. The algebraic steps are verified and the logic is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly integrates the differential equation and applies the initial condition to determine the constant of integration. The algebraic steps are verified and the logic is sound.gpt-oss:20b: fail (misleading) 2026-10-18 — Step 2 incorrectly states "8 - Subs(x**3/3 - 2*x, x, 0) = 8"; the correct reasoning is that y(0)=C, so C=8. The expression as written is misleading and could confuse a student.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the general antiderivative and determines the constant of integration using the initial condition. The steps are logically sound and algebraically verified.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the general antiderivative and determines the constant of integration using the initial condition. The logic is sound and the steps are valid.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution never actually solves for the constant of integration C. Step 2 merely shows that 8−0=8, which does not determine C, yet step 3 proceeds as if C=8. This misleads a student into thinking the constant was correctly found when it was not.
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.