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Antiderivatives with initial conditions

Problem 3.626 · hard

Solve the initial-value problem \( \displaystyle f'(x) = 4 \sqrt{x} + 4 x^{3} \), \( \displaystyle f(1) = -1 \).

\[ \frac{d}{d x} \left(\frac{8 x^{\frac{3}{2}}}{3} + x^{4}\right) = 4 \sqrt{x} + 4 x^{3} \]
An antiderivative; every other differs by a constant C.✓ Proved
\[ - \left. x^{4} + \frac{8 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} - 1 = - \frac{14}{3} \]
f(1) = -1 fixes C.✓ Proved
\[ x^{4} + \frac{8 x^{\frac{3}{2}}}{3} - \frac{14}{3} = \frac{8 x^{\frac{3}{2}}}{3} + x^{4} - \frac{14}{3} \]
The solution.✓ Proved

Answer \( f(x) = \frac{8 x^{\frac{3}{2}}}{3} + x^{4} - \frac{14}{3} \)

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 second sentence incorrectly applies the initial condition. It subtracts 1 from the antiderivative evaluated at 1 and sets the result to -14/3, which does not correctly solve for the constant of integration. The correct approach is to evaluate the antiderivative at x=1, set it equal to -1, and solve for the constant. This step is essential for obtaining the correct particular solution.
qwen3.6:27b-mlx: pass — The solution correctly identifies the general antiderivative and uses the initial condition to determine the constant of integration. The logic is sound and the final answer is correct.

Every verdict on record (4)

gpt-oss:20b: fail (error) 2026-10-13 — The second sentence incorrectly applies the initial condition. It subtracts 1 from the antiderivative evaluated at 1 and sets the result to -14/3, which does not correctly solve for the constant of integration. The correct approach is to evaluate the antiderivative at x=1, set it equal to -1, and solve for the constant. This step is essential for obtaining the correct particular solution.
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the general antiderivative and uses the initial condition to determine the constant of integration. The logic is sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13

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 2026-10-13 with SymPy 1.14.0.