Antiderivatives with initial conditions
Problem 3.633 · hard
Solve \( \displaystyle f''(x) = - 3 x^{2} - 4 \cos{\left(x \right)} \) with \( \displaystyle f'(0) = -5 \) and \( \displaystyle f(0) = 4 \).
- \[ \frac{d}{d x} \left(- x^{3} - 4 \sin{\left(x \right)}\right) = - 3 x^{2} - 4 \cos{\left(x \right)} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ - \left. - x^{3} - 4 \sin{\left(x \right)} \right|_{\substack{ x=0 }} - 5 = -5 \]f′(0) = -5 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{4}}{4} - 5 x + 4 \cos{\left(x \right)}\right) = - x^{3} - 4 \sin{\left(x \right)} - 5 \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 4 - \left. - 5 x + \frac{\left(-1\right) x^{4}}{4} + 4 \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f(0) = 4 fixes C₀.✓ Proved
- \[ - 5 x + \frac{\left(-1\right) x^{4}}{4} + 4 \cos{\left(x \right)} = - \frac{x^{4}}{4} - 5 x + 4 \cos{\left(x \right)} \]The solution.✓ Proved
Answer \( f(x) = - \frac{x^{4}}{4} - 5 x + 4 \cos{\left(x \right)} \)
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: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution incorrectly integrates the term -4 cos(x) as +4 cos(x) instead of -4 sin(x). Consequently, the derivative of the stated answer does not match the required f'(x), and the second derivative does not match f''(x).
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.