Antiderivatives with initial conditions
Problem 3.383 · hard
Solve \( \displaystyle f''(x) = 2 - \frac{1}{x^{2}} \) with \( \displaystyle f'(1) = 1 \) and \( \displaystyle f(1) = -5 \).
- \[ \frac{d}{d x} \left(2 x + \frac{1}{x}\right) = 2 - \frac{1}{x^{2}} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ 1 - \left. 2 x + 1 \frac{1}{x} \right|_{\substack{ x=1 }} = -2 \]f′(1) = 1 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(x^{2} - 2 x + \ln{\left(x \right)}\right) = 2 x - 2 + \frac{1}{x} \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ - \left. x^{2} - 2 x + \ln{\left(x \right)} \right|_{\substack{ x=1 }} - 5 = -4 \]f(1) = -5 fixes C₀.✓ Proved
- \[ x^{2} - 2 x + \ln{\left(x \right)} - 4 \]The solution.✓ Proved
Answer \( f(x) = x^{2} - 2 x + \ln{\left(x \right)} - 4 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 antiderivatives and applies the initial conditions to determine the constants of integration. The final function satisfies the differential equation and boundary conditions.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the antiderivatives and applies the initial conditions to determine the constants of integration. The final function satisfies the differential equation and boundary conditions.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates f''(x) to find f'(x), applies the initial condition f'(1)=1 to determine the constant, integrates again to find f(x), and applies f(1)=-5 to find the final constant. The algebraic steps and logic are sound.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.