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Second-order linear equations

Problem 6.135 · medium

Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = 1 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
\[ \left(m + 2\right)^{2} = m^{2} + 4 m + 4 \]
Roots: -2.✓ Proved
A repeated root r gives y = (C₁ + C₂x)e^(rx).
Use y(0) and y'(0) to find C₁ and C₂.
\[ 4 x e^{- 2 x} + 4 \frac{d}{d x} x e^{- 2 x} + \frac{d^{2}}{d x^{2}} x e^{- 2 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ 0 \]
y(0).✓ Proved
\[ 1 \]
y'(0).✓ Proved

Answer \( y = x e^{- 2 x} \)

Lines: 4 proved, 3 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.

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
4	Not checked	—	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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly uses \(y=4x e^{-2x}\) instead of the correct \(y=x e^{-2x}\). The constants \(C_1\) and \(C_2\) are never actually solved for, an
qwen3.6:27b-mlx: fail (error) — The solution fails to actually solve for the constants C1 and C2. It states the general form and then verifies that the final answer satisfies the ODE and initial conditions, but it skips the logical step of deriving the specific constants from the initial conditions.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to actually solve for the constants C1 and C2. It states the general form and then verifies that the final answer satisfies the ODE and initial conditions, but it skips the logical step of deriving the specific constants from the initial conditions.
gpt-oss:20b: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly uses \(y=4x e^{-2x}\) instead of the correct \(y=x e^{-2x}\). The constants \(C_1\) and \(C_2\) are never actually solved for, an
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly determine the constants C1 and C2. It jumps from the general form to the final answer without showing the calculation that yields C1=0 and C2=1, making the derivation incomplete.
gpt-oss:20b: fail (style) 2026-09-27 — The solution claims to determine the constants from the initial conditions but never actually solves for C₁ and C₂, nor explicitly states the final form y=x e^(–2x). The steps are incomplete, making the explanation unclear.

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 2026-09-27 with SymPy 1.14.0.