Differential equations by direct integration
Problem 6.431 · medium
Find the general solution of \( \displaystyle y'' = \sin{\left(2 x \right)} \).
- \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x \right)}}{2}\right) = \sin{\left(2 x \right)} \]Integrate once: y′ = -cos(2*x)/2 + C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{\sin{\left(2 x \right)}}{4}\right) = - \frac{\cos{\left(2 x \right)}}{2} \]Integrate again: y = -sin(2*x)/4 + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} - \frac{\sin{\left(2 x \right)}}{4} \)
✓ 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. How we verify
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | finite differences of the answer match f, and the initial value holds |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of successive integration to solve the second-order differential equation. The algebraic steps and the final general solution are correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14
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/de_direct_integration, checked 2026-10-17 with SymPy 1.14.0.