Antiderivatives with initial conditions
Problem 3.531 · hard
Solve \( \displaystyle f''(x) = - 2 x^{2} + \frac{1}{x^{2}} \) with \( \displaystyle f'(1) = 0 \) and \( \displaystyle f(1) = 3 \).
- \[ \frac{d}{d x} \left(- \frac{2 x^{3}}{3} - \frac{1}{x}\right) = - 2 x^{2} + \frac{1}{x^{2}} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ - \left. - \frac{2 x^{3}}{3} - 1 \frac{1}{x} \right|_{\substack{ x=1 }} = \frac{5}{3} \]f′(1) = 0 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{4}}{6} + \frac{5 x}{3} - \ln{\left(x \right)}\right) = - \frac{2 x^{3}}{3} + \frac{5}{3} - \frac{1}{x} \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 3 - \left. \frac{5 x}{3} + \frac{\left(-1\right) x^{4}}{6} - \ln{\left(x \right)} \right|_{\substack{ x=1 }} = \frac{3}{2} \]f(1) = 3 fixes C₀.✓ Proved
- \[ \frac{5 x}{3} + \frac{\left(-1\right) x^{4}}{6} - \ln{\left(x \right)} + \frac{3}{2} = - \frac{x^{4}}{6} + \frac{5 x}{3} - \ln{\left(x \right)} + \frac{3}{2} \]The solution.✓ Proved
Answer \( f(x) = - \frac{x^{4}}{6} + \frac{5 x}{3} - \ln{\left(x \right)} + \frac{3}{2} \)
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 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09gpt-oss:20b: fail (misleading) 2026-10-09 — Step 2 incorrectly states that Subs(-2*x**3/3 - 1/x, x, 1) equals 5/3; the correct value is -5/3. The subsequent conclusion is correct, but the sentence misleads a reader about the evaluation.qwen3.6:27b-mlx: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.