Variation of parameters
Problem 6.514 · hard
Use variation of parameters to find a particular solution of \( \displaystyle y + y'' = \frac{1}{\sin{\left(x \right)}} \), given that \( \displaystyle y_1 = \cos{\left(x \right)} \) and \( \displaystyle y_2 = \sin{\left(x \right)} \) solve the homogeneous equation.
- \[ - \sin{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} \sin{\left(x \right)} = 1 \]The Wronskian W = y₁y₂′ − y₂y₁′.✓ Proved
- \[ \frac{d}{d x} \left(- x\right) = -1 \]u₁′ = −y₂g/W; u₁ = -x.✓ Proved
- \[ \frac{d}{d x} \ln{\left(\sin{\left(x \right)} \right)} = \frac{1}{\tan{\left(x \right)}} \]u₂′ = y₁g/W; u₂ = log(sin(x)).✓ Proved
- y_p = u₁y₁ + u₂y₂: -x*cos(x) + log(sin(x))*sin(x).Reviewed
- \[ - x \cos{\left(x \right)} + \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(- x \cos{\left(x \right)} + \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \frac{1}{\sin{\left(x \right)}} \]It satisfies the equation.✓ Proved
Answer \( y_p = - x \cos{\left(x \right)} + \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | finite differences of the answer satisfy the equation at three points |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the method of variation of parameters, including the correct calculation of the Wronskian and the integrals for u1 and u2. The final particular solution is verified to satisfy the differential equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the method of variation of parameters, including the correct calculation of the Wronskian and the integrals for u1 and u2. The final particular solution is verified to satisfy the differential equation.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the method of variation of parameters, including the correct calculation of the Wronskian and the integration of the parameter derivatives. The final particular solution is verified to satisfy the differential equation.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of variation of parameters. The Wronskian is calculated correctly, the integrals for u1 and u2 are performed correctly, and the final particular solution satisfies the differential equation.gpt-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/variation_of_parameters, checked 2026-10-17 with SymPy 1.14.0.