Home›Calculus 3›Second-order linear equations›Problem 6.113

Second-order linear equations

Problem 6.113 · hard

Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -1 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
Reviewed
\[ \left(m + 3\right)^{2} = m^{2} + 6 m + 9 \]
Roots: -3.✓ Proved
A repeated root r gives y = (C₁ + C₂x)e^(rx).
Reviewed
Use y(0) and y'(0) to find C₁ and C₂.
Reviewed
\[ \left(- 36 x - 9\right) e^{- 3 x} + 6 \frac{d}{d x} \left(- 4 x - 1\right) e^{- 3 x} + \frac{d^{2}}{d x^{2}} \left(- 4 x - 1\right) e^{- 3 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
y(0).✓ Proved
\[ -1 \]
y'(0).✓ Proved

Answer \( y = \left(- 4 x - 1\right) e^{- 3 x} \)

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, states the general form, and verifies the specific solution against the differential equation and initial conditions.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, states the general form, and verifies the specific solution against the differential equation and initial conditions.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions, jumping directly to the final answer without showing the necessary algebraic steps. This makes the solution incomplete and does not demonstrate the method for finding the specific solution.
gpt-oss:20b: pass 2026-09-26

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.