Antiderivatives with initial conditions
Problem 3.379 · hard
Solve \( \displaystyle f''(x) = - 2 \sqrt{x} + 4 x^{2} \) with \( \displaystyle f'(1) = 4 \) and \( \displaystyle f(1) = 2 \).
- \[ \frac{d}{d x} \left(- \frac{4 x^{\frac{3}{2}}}{3} + \frac{4 x^{3}}{3}\right) = - 2 \sqrt{x} + 4 x^{2} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ 4 - \left. \frac{4 x^{3}}{3} - \frac{4 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} = 4 \]f′(1) = 4 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{8 x^{\frac{5}{2}}}{15} + \frac{x^{4}}{3} + 4 x\right) = - \frac{4 x^{\frac{3}{2}}}{3} + \frac{4 x^{3}}{3} + 4 \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ 2 - \left. \frac{x^{4}}{3} + 4 x - \frac{8 x^{\frac{5}{2}}}{15} \right|_{\substack{ x=1 }} = - \frac{9}{5} \]f(1) = 2 fixes C₀.✓ Proved
- \[ \frac{x^{4}}{3} + 4 x - \frac{8 x^{\frac{5}{2}}}{15} - \frac{9}{5} = - \frac{8 x^{\frac{5}{2}}}{15} + \frac{x^{4}}{3} + 4 x - \frac{9}{5} \]The solution.✓ Proved
Answer \( f(x) = - \frac{8 x^{\frac{5}{2}}}{15} + \frac{x^{4}}{3} + 4 x - \frac{9}{5} \)
✓ 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)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates term-by-term and applies the initial conditions to determine the constants of integration. The algebraic steps are verified and the final function satisfies the differential equation and boundary conditions.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.