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Second-order linear equations

Problem 6.109 · hard

Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -2 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
Reviewed
\[ \left(m + 2\right)^{2} = m^{2} + 4 m + 4 \]
Roots: -2.✓ Proved
A repeated root r gives y = (C₁ + C₂x)e^(rx).
Reviewed
Use y(0) and y'(0) to find C₁ and C₂.
Reviewed
\[ \left(- 24 x - 8\right) e^{- 2 x} + 4 \frac{d}{d x} \left(- 6 x - 2\right) e^{- 2 x} + \frac{d^{2}}{d x^{2}} \left(- 6 x - 2\right) e^{- 2 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ -2 \]
y(0).✓ Proved
\[ -2 \]
y'(0).✓ Proved

Answer \( y = 2 \left(- 3 x - 1\right) e^{- 2 x} \)

Lines: 4 proved, 3 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.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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
6	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
7	✓ 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	sympy.checkodesol substitutes the solution back; both initial conditions match

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. Although the constants C1 and C2 are not explicitly solved for in the text, the final answer is derived correctly and verified.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. Although the constants C1 and C2 are not explicitly solved for in the text, the final answer is derived correctly and verified.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. The steps are logically sound and mathematically correct.
gpt-oss:20b: fail (misleading) 2026-09-26 — The solution skips the crucial step of determining the constants C₁ and C₂ from the initial conditions and presents an incorrect intermediate expression (8*(-3*x-1)*exp(-2*x)) that does not match the claimed final solution. This would mislead a student into believing the verification step is correct without proper justification.

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/second_order_linear, checked 2026-09-26 with SymPy 1.14.0.