Antiderivatives with initial conditions
Problem 3.576 · medium
Solve the initial-value problem \( \displaystyle f'(x) = x^{2} - \frac{4}{x^{2}} \), \( \displaystyle f(1) = 2 \).
- \[ \frac{d}{d x} \left(\frac{x^{3}}{3} + \frac{4}{x}\right) = x^{2} - \frac{4}{x^{2}} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ 2 - \left. \frac{x^{3}}{3} + \frac{4}{x} \right|_{\substack{ x=1 }} = - \frac{7}{3} \]f(1) = 2 fixes C.✓ Proved
- \[ \frac{x^{3}}{3} - \frac{7}{3} + \frac{4}{x} \]The solution.✓ Proved
Answer \( f(x) = \frac{x^{3}}{3} - \frac{7}{3} + \frac{4}{x} \)
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 |
| 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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the antiderivative of -4/x^2 as 4/x. The derivative of 4/x is -4/x^2, so the term should be +4/x in the antideri
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the antiderivative of -4/x^2 as 4/x. The derivative of 4/x is -4/x^2, so the term should be +4/x in the antiderigpt-oss:20b: pass 2026-10-11gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The antiderivative of -4/x^2 is 4/x, but the constant C is calculated incorrectly. f(1) = 1/3 + 4 + C = 2 implies C = -4/3, not -7/3. The stated answer yields f(1) = 1/3 - 7/3 + 4 = 8/3, which contradicts the initial condition f(1)=2.
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.