Antiderivatives with initial conditions
Problem 3.584 · hard
Solve the initial-value problem \( \displaystyle f'(x) = - 2 \sqrt{x} + 4 x \), \( \displaystyle f(1) = -3 \).
- \[ \frac{d}{d x} \left(- \frac{4 x^{\frac{3}{2}}}{3} + 2 x^{2}\right) = - 2 \sqrt{x} + 4 x \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ - \left. 2 x^{2} - \frac{4 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} - 3 = - \frac{11}{3} \]f(1) = -3 fixes C.✓ Proved
- \[ 2 x^{2} - \frac{4 x^{\frac{3}{2}}}{3} - \frac{11}{3} = - \frac{4 x^{\frac{3}{2}}}{3} + 2 x^{2} - \frac{11}{3} \]The solution.✓ Proved
Answer \( f(x) = - \frac{4 x^{\frac{3}{2}}}{3} + 2 x^{2} - \frac{11}{3} \)
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: fail (error) — The constant of integration is omitted. Evaluating the antiderivative at x=1 gives 2/3, so to satisfy f(1)=-3 we need C=-11/3, not the value computed in step 2.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11gpt-oss:20b: fail (error) 2026-10-11 — The constant of integration is omitted. Evaluating the antiderivative at x=1 gives 2/3, so to satisfy f(1)=-3 we need C=-11/3, not the value computed in step 2.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the antiderivative, solves for the constant of integration using the initial condition, and presents the correct final function.gpt-oss:20b: pass 2026-10-11
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.