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Variation of parameters

Problem 6.505 · hard

Use variation of parameters to find a particular solution of \( \displaystyle 4 y + y'' = \frac{3}{\cos{\left(2 x \right)}} \), 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
\[ \frac{d}{d x} \frac{3 \ln{\left(\cos{\left(2 x \right)} \right)}}{4} = - \frac{3 \tan{\left(2 x \right)}}{2} \]
u₁′ = −y₂g/W; u₁ = 3*log(cos(2*x))/4.✓ Proved
\[ \frac{d}{d x} \frac{3 x}{2} = \frac{3}{2} \]
u₂′ = y₁g/W; u₂ = 3*x/2.✓ Proved
y_p = u₁y₁ + u₂y₂: 3*x*sin(2*x)/2 + 3*log(cos(2*x))*cos(2*x)/4.
\[ 6 x \sin{\left(2 x \right)} + 3 \ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)} + \frac{d^{2}}{d x^{2}} \left(\frac{3 x \sin{\left(2 x \right)}}{2} + \frac{3 \ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}}{4}\right) = \frac{3}{\cos{\left(2 x \right)}} \]
It satisfies the equation.✓ Proved

Answer \( y_p = \frac{3 x \sin{\left(2 x \right)}}{2} + \frac{3 \ln{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}}{4} \)

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: fail (error) — The standard form of the ODE requires dividing by the leading coefficient 4, so g(x) should be 3/(4 cos(2x)). The solution uses g(x) = 3/cos(2x), resulting in Wronskian-based integrals that are off by a factor of 4. Although the final answer is correct, the intermediate steps for u1' and u2' are mathematically inconsistent with the stated method and the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The standard form of the ODE requires dividing by the leading coefficient 4, so g(x) should be 3/(4 cos(2x)). The solution uses g(x) = 3/cos(2x), resulting in Wronskian-based integrals that are off by a factor of 4. Although the final answer is correct, the intermediate steps for u1' and u2' are mathematically inconsistent with the stated method and the problem statement.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the method of variation of parameters, including the necessary step of dividing the non-homogeneous term by the leading coefficient (4) to find g(x). The Wronskian, integrals for u1 and u2, and final assembly are all correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The Wronskian calculation in step 1 is incorrect; it yields 2 instead of the correct value 4. Consequently, the integrals for u1 and u2 in steps 2 and 3 are off by a factor of 2, leading to an incorrect particular solution.
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.