Antiderivatives with initial conditions
Problem 3.527 · hard
Solve \( \displaystyle f''(x) = 2 \sqrt{x} - 3 x^{2} \) with \( \displaystyle f'(1) = -4 \) and \( \displaystyle f(1) = 2 \).
- \[ \frac{d}{d x} \left(\frac{4 x^{\frac{3}{2}}}{3} - x^{3}\right) = 2 \sqrt{x} - 3 x^{2} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ - \left. - x^{3} + \frac{4 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} - 4 = - \frac{13}{3} \]f′(1) = -4 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(\frac{8 x^{\frac{5}{2}}}{15} - \frac{x^{4}}{4} - \frac{13 x}{3}\right) = \frac{4 x^{\frac{3}{2}}}{3} - x^{3} - \frac{13}{3} \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 2 - \left. - \frac{x^{4}}{4} - \frac{13 x}{3} + \frac{8 x^{\frac{5}{2}}}{15} \right|_{\substack{ x=1 }} = \frac{121}{20} \]f(1) = 2 fixes C₀.✓ Proved
- \[ - \frac{x^{4}}{4} - \frac{13 x}{3} + \frac{8 x^{\frac{5}{2}}}{15} + \frac{121}{20} = \frac{8 x^{\frac{5}{2}}}{15} - \frac{x^{4}}{4} - \frac{13 x}{3} + \frac{121}{20} \]The solution.✓ Proved
Answer \( f(x) = \frac{8 x^{\frac{5}{2}}}{15} - \frac{x^{4}}{4} - \frac{13 x}{3} + \frac{121}{20} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: pass 2026-10-09qwen3.6:27b-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.