Antiderivatives with initial conditions
Problem 3.625 · hard
Solve \( \displaystyle f''(x) = 6 x^{2} - \sin{\left(x \right)} \) with \( \displaystyle f'(0) = 1 \) and \( \displaystyle f(0) = 6 \).
- \[ \frac{d}{d x} \left(2 x^{3} + \cos{\left(x \right)}\right) = 6 x^{2} - \sin{\left(x \right)} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ 1 - \left. 2 x^{3} + \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f′(0) = 1 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(\frac{x^{4}}{2} + \sin{\left(x \right)}\right) = 2 x^{3} + \cos{\left(x \right)} \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 6 - \left. \frac{x^{4}}{2} + \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 6 \]f(0) = 6 fixes C₀.✓ Proved
- \[ \frac{x^{4}}{2} + \sin{\left(x \right)} + 6 \]The solution.✓ Proved
Answer \( f(x) = \frac{x^{4}}{2} + \sin{\left(x \right)} + 6 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the final function f(x) including the integration constants determined in steps 2 and 4. While step 5 shows an identity, it does not clearly present f(x) = x^4/2 + sin(x) + 6 as the result of combining the antiderivatives with the calculated constants C1=0 and C0=6. The logical flow from the constants to the final expression is missing.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to explicitly state the final function f(x) including the integration constants determined in steps 2 and 4. While step 5 shows an identity, it does not clearly present f(x) = x^4/2 + sin(x) + 6 as the result of combining the antiderivatives with the calculated constants C1=0 and C0=6. The logical flow from the constants to the final expression is missing.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly integrates twice and applies the initial conditions to determine the constants of integration. The logic is sound and the final answer is correct.
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 2026-10-13 with SymPy 1.14.0.