Home›Calculus 3›Variation of parameters›Problem 6.646

Variation of parameters

Problem 6.646 · hard

Use variation of parameters to find a particular solution of \( \displaystyle y + y'' = \frac{3}{\sin{\left(x \right)}} \), 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
\[ \frac{d}{d x} \left(- 3 x\right) = -3 \]
u₁′ = −y₂g/W; u₁ = -3*x.✓ Proved
\[ \frac{d}{d x} 3 \ln{\left(\sin{\left(x \right)} \right)} = \frac{3}{\tan{\left(x \right)}} \]
u₂′ = y₁g/W; u₂ = 3*log(sin(x)).✓ Proved
y_p = u₁y₁ + u₂y₂: -3*x*cos(x) + 3*log(sin(x))*sin(x).
\[ - 3 x \cos{\left(x \right)} + 3 \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(- 3 x \cos{\left(x \right)} + 3 \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \frac{3}{\sin{\left(x \right)}} \]
It satisfies the equation.✓ Proved

Answer \( y_p = - 3 x \cos{\left(x \right)} + 3 \ln{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \)

Lines: 4 proved, 1 not checked. 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	Not checked	—	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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims u_2' = 3/tan(x) (which is 3 cot(x)), but the correct derivative for u_2 = 3 ln(sin(x)) is 3 cot(x). However, the formula for u_2

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims u_2' = 3/tan(x) (which is 3 cot(x)), but the correct derivative for u_2 = 3 ln(sin(x)) is 3 cot(x). However, the formula for u_2
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the method of variation of parameters, with correct Wronskian calculation, correct integrals for u1 and u2, and correct final assembly of the particular solution.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the method of variation of parameters, including the correct calculation of the Wronskian, the integrals for u1 and u2, and the final assembly of the particular solution.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.