Antiderivatives with initial conditions
Problem 3.575 · medium
Solve the initial-value problem \( \displaystyle f'(x) = 2 x^{2} - e^{x} \), \( \displaystyle f(0) = -4 \).
- \[ \frac{d}{d x} \left(\frac{2 x^{3}}{3} - e^{x}\right) = 2 x^{2} - e^{x} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ - \left. \frac{2 x^{3}}{3} - e^{x} \right|_{\substack{ x=0 }} - 4 = -3 \]f(0) = -4 fixes C.✓ Proved
- \[ \frac{2 x^{3}}{3} - e^{x} - 3 \]The solution.✓ Proved
Answer \( f(x) = \frac{2 x^{3}}{3} - e^{x} - 3 \)
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) — Step 2 incorrectly derives the constant: it uses "-Subs(...) - 4 = -3" instead of the correct equation Subs(...) + C = -4, even though the final constant happens to be correct. This misleads a student about how to apply the initial condition.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) 2026-10-11 — Step 2 incorrectly derives the constant: it uses "-Subs(...) - 4 = -3" instead of the correct equation Subs(...) + C = -4, even though the final constant happens to be correct. This misleads a student about how to apply the initial condition.qwen3.6:27b-mlx: pass 2026-10-11 — 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 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the antiderivative and determines the constant of integration using the initial condition. The steps are logically sound and algebraically 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.