Home›Calculus 3›Variation of parameters›Problem 6.510

Variation of parameters

Problem 6.510 · hard

Use variation of parameters to find a particular solution of \( \displaystyle 4 y + y'' = \frac{2}{\cos{\left(2 x \right)}} \), given that \( \displaystyle y_1 = \cos{\left(2 x \right)} \) and \( \displaystyle y_2 = \sin{\left(2 x \right)} \) solve the homogeneous equation.

\[ - \sin{\left(2 x \right)} \frac{d}{d x} \cos{\left(2 x \right)} + \cos{\left(2 x \right)} \frac{d}{d x} \sin{\left(2 x \right)} = 2 \]
The Wronskian W = y₁y₂′ − y₂y₁′.✓ Proved
\[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} = - \tan{\left(2 x \right)} \]
u₁′ = −y₂g/W; u₁ = log(cos(2*x))/2.✓ Proved
\[ \frac{d}{d x} x = 1 \]
u₂′ = y₁g/W; u₂ = x.✓ Proved
y_p = u₁y₁ + u₂y₂: x*sin(2*x) + log(cos(2*x))*cos(2*x)/2.
Reviewed
\[ 4 x \sin{\left(2 x \right)} + 2 \ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)} + \frac{d^{2}}{d x^{2}} \left(x \sin{\left(2 x \right)} + \frac{\ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}}{2}\right) = \frac{2}{\cos{\left(2 x \right)}} \]
It satisfies the equation.✓ Proved

Answer \( y_p = x \sin{\left(2 x \right)} + \frac{\ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}}{2} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the method of variation of parameters, including the necessary division by the leading coefficient to standardize the differential equation before applying the formulas for u1' and u2'. The final result is verified.

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 necessary division by the leading coefficient to standardize the differential equation before applying the formulas for u1' and u2'. The final result is verified.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to normalize the differential equation to standard form (leading coefficient 1) before applying the variation of parameters formulas. The forcing term g(x) should be 1/(2 cos(2x)), not 2/cos(2x), which results in incorrect particular solution coefficients.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the method of variation of parameters, including the necessary division by the leading coefficient 4 to identify the forcing function g(x). The Wronskian, integrals for u1 and u2, and final verification are all correct.
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.