Antiderivatives with initial conditions
Problem 3.577 · hard
Solve the initial-value problem \( \displaystyle f'(x) = - 3 \sqrt{x} + 4 x^{2} \), \( \displaystyle f(1) = -5 \).
- \[ \frac{d}{d x} \left(- 2 x^{\frac{3}{2}} + \frac{4 x^{3}}{3}\right) = - 3 \sqrt{x} + 4 x^{2} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ - \left. \frac{4 x^{3}}{3} - 2 x^{\frac{3}{2}} \right|_{\substack{ x=1 }} - 5 = - \frac{13}{3} \]f(1) = -5 fixes C.✓ Proved
- \[ \frac{4 x^{3}}{3} - 2 x^{\frac{3}{2}} - \frac{13}{3} = - 2 x^{\frac{3}{2}} + \frac{4 x^{3}}{3} - \frac{13}{3} \]The solution.✓ Proved
Answer \( f(x) = - 2 x^{\frac{3}{2}} + \frac{4 x^{3}}{3} - \frac{13}{3} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and states the final result. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and states the final result. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-11gpt-oss:20b: fail (error) 2026-10-11 — The constant C is computed incorrectly: f(1)= -2/3 + 4/3 -5 = -17/3, not -13/3. The solution should be f(x)= -2x^{3/2} + frac{4}{3}x^3 - frac{17}{3}.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 final function. The logic is sound and the algebra is correct.
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.