Antiderivatives with initial conditions
Problem 3.579 · hard
Solve \( \displaystyle f''(x) = - 6 x^{2} + \sin{\left(x \right)} \) with \( \displaystyle f'(0) = 0 \) and \( \displaystyle f(0) = 4 \).
- \[ \frac{d}{d x} \left(- 2 x^{3} - \cos{\left(x \right)}\right) = - 6 x^{2} + \sin{\left(x \right)} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ - \left. - 2 x^{3} - \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]f′(0) = 0 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{4}}{2} + x - \sin{\left(x \right)}\right) = - 2 x^{3} - \cos{\left(x \right)} + 1 \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 4 - \left. x + \frac{\left(-1\right) x^{4}}{2} - \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 4 \]f(0) = 4 fixes C₀.✓ Proved
- \[ x + \frac{\left(-1\right) x^{4}}{2} - \sin{\left(x \right)} + 4 = - \frac{x^{4}}{2} + x - \sin{\left(x \right)} + 4 \]The solution.✓ Proved
Answer \( f(x) = - \frac{x^{4}}{2} + x - \sin{\left(x \right)} + 4 \)
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.
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | the answer differentiated back to the given derivative, and the initial values checked |
Reviewers
gpt-oss:20b: fail (misleading) — The solution’s sentences about determining the integration constants are unclear and incorrectly presented. Step 2 uses a negative sign and omits the constant C₁, while step 4 similarly skips the explicit introduction of C₀. A student could be confused about how the constants are obtained from the initial conditions.qwen3.6:27b-mlx: pass — The solution correctly integrates twice and applies the initial conditions to determine the constants of integration. The logic is sound and the final answer matches the problem statement.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-11 — The solution’s sentences about determining the integration constants are unclear and incorrectly presented. Step 2 uses a negative sign and omits the constant C₁, while step 4 similarly skips the explicit introduction of C₀. A student could be confused about how the constants are obtained from the initial conditions.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly integrates twice and applies the initial conditions to determine the constants of integration. The logic is sound and the final answer matches the problem statement.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly integrates twice and applies the initial conditions to determine the constants of integration. The final result matches the stated answer.
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/antiderivative_ivp, checked 2026-10-11 with SymPy 1.14.0.