Home›Calculus 3›Variation of parameters›Problem 6.501

Variation of parameters

Problem 6.501 · hard

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

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

Answer \( y_p = 2 \left(- \ln{\left(x \right)} - 1\right) e^{x} \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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. The Wronskian is calculated correctly, the integrals for u1 and u2 are performed accurately, and the final particular solution 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. The Wronskian is calculated correctly, the integrals for u1 and u2 are performed accurately, and the final particular solution satisfies the differential equation.
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 correct calculation of the Wronskian and the integrals for the parameters u1 and u2. The final particular solution is verified to satisfy the differential equation.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the non-homogeneous term g(x) as 2e^x/x^2 instead of 2/x^2. The standard form requires the coefficient of y'' to be 1, so g(x) is the RHS divided by 1. The Wronskian calculation is correct, but the subsequent integrals for u1' and u2' use the wrong g(x), 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.