Antiderivatives with initial conditions
Problem 3.382 · hard
Solve \( \displaystyle f''(x) = 4 \cos{\left(x \right)} + 4 \) with \( \displaystyle f'(0) = 3 \) and \( \displaystyle f(0) = -4 \).
- \[ \frac{d}{d x} \left(4 x + 4 \sin{\left(x \right)}\right) = 4 \cos{\left(x \right)} + 4 \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ 3 - \left. 4 x + 4 \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 3 \]f′(0) = 3 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{2} + 3 x - 4 \cos{\left(x \right)}\right) = 4 x + 4 \sin{\left(x \right)} + 3 \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ - \left. 2 x^{2} + 3 x - 4 \cos{\left(x \right)} \right|_{\substack{ x=0 }} - 4 = 0 \]f(0) = -4 fixes C₀.✓ Proved
- \[ 2 x^{2} + 3 x - 4 \cos{\left(x \right)} \]The solution.✓ Proved
Answer \( f(x) = 2 x^{2} + 3 x - 4 \cos{\left(x \right)} \)
✓ 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 integrates twice and applies the initial conditions to determine the constants. The logic is sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates twice and applies the initial conditions to determine the constants. The logic is sound and the final answer is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-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.