Variation of parameters
Problem 6.508 · hard
Use variation of parameters to find a particular solution of \( \displaystyle y + y'' = \frac{2}{\cos{\left(x \right)}} \), 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
- \[ \frac{d}{d x} 2 \ln{\left(\cos{\left(x \right)} \right)} = - 2 \tan{\left(x \right)} \]u₁′ = −y₂g/W; u₁ = 2*log(cos(x)).✓ Proved
- \[ \frac{d}{d x} 2 x = 2 \]u₂′ = y₁g/W; u₂ = 2*x.✓ Proved
- y_p = u₁y₁ + u₂y₂: 2*x*sin(x) + 2*log(cos(x))*cos(x).Reviewed
- \[ 2 x \sin{\left(x \right)} + 2 \ln{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(2 x \sin{\left(x \right)} + 2 \ln{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) = \frac{2}{\cos{\left(x \right)}} \]It satisfies the equation.✓ Proved
Answer \( y_p = 2 x \sin{\left(x \right)} + 2 \ln{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \)
Lines: 4 proved, 1 reviewed. 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. 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, deriving the correct particular solution and verifying it satisfies 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, deriving the correct particular solution and verifying it satisfies the differential equation.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to explicitly state the standard form of the differential equation (leading coefficient 1) and the formula for variation of paramgpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of variation of parameters, including the correct calculation of the Wronskian, the integrals for the parameters u1 and u2, and the final assembly of the particular solution.gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The Wronskian is computed incorrectly: it should be \(W=y_1y_2'-y_2y_1'=\cos^2x+\sin^2x=1\), not \(-\sin^2x+\cos^2x\). This mistake propagates to the form
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.