Antiderivatives with initial conditions
Problem 3.630 · hard
Solve the initial-value problem \( \displaystyle f'(x) = - 4 x + \frac{3}{x^{3}} \), \( \displaystyle f(1) = -4 \).
- \[ \frac{d}{d x} \left(- 2 x^{2} - \frac{3}{2 x^{2}}\right) = - 4 x + \frac{3}{x^{3}} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ - \left. - 2 x^{2} - \frac{3}{2 x^{2}} \right|_{\substack{ x=1 }} - 4 = - \frac{1}{2} \]f(1) = -4 fixes C.✓ Proved
- \[ - 2 x^{2} - 1 \cdot \frac{1}{2} - \frac{3}{2 x^{2}} = - 2 x^{2} - \frac{1}{2} - \frac{3}{2 x^{2}} \]The solution.✓ Proved
Answer \( f(x) = - 2 x^{2} - \frac{1}{2} - \frac{3}{2 x^{2}} \)
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 (misleading) — The solution skips the step of adding an arbitrary constant to the antiderivative before applying the initial condition, and the wording in step 2 is confusing. A student could misunderstand how the constant is determined.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative, applies the initial condition to determine the constant of integration, and presents the correct final function.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-13 — The solution skips the step of adding an arbitrary constant to the antiderivative before applying the initial condition, and the wording in step 2 is confusing. A student could misunderstand how the constant is determined.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the antiderivative, applies the initial condition to determine the constant of integration, and presents the correct final function.gpt-oss:20b: fail (misleading) 2026-10-13 — The solution’s second sentence incorrectly derives the constant: it subtracts 4 from the antiderivative instead of adding a constant term. This misleads a student about how to apply the initial condition. The final expression is correct, but the reasoning is flawed.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the antiderivative and applies the initial condition to determine the constant of integration. The algebraic steps are correct and the final answer matches the stated answer.
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.