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Second-order linear equations

Problem 6.186 · medium

Solve \( \displaystyle y'' - 1y' - 6y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = 2 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
\[ \left(m - 3\right) \left(m + 2\right) = m^{2} - m - 6 \]
Roots: 3, -2.✓ Proved
Two real roots give y = C₁e^(r₁x) + C₂e^(r₂x).
Use y(0) and y'(0) to find C₁ and C₂.
\[ - \frac{d}{d x} \left(- e^{- 2 x}\right) + \frac{d^{2}}{d x^{2}} \left(- e^{- 2 x}\right) + 6 e^{- 2 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ -1 \]
y(0).✓ Proved
\[ 2 \]
y'(0).✓ Proved

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

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: pass
qwen3.6:27b-mlx: fail (error) — The solution claims the answer is -exp(-2*x) but never solves for the constants C1 and C2 using the initial conditions. It simply verifies that the proposed answer satisfies the ODE and initial conditions, skipping the actual derivation step implied by lines 3 and 4.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution claims the answer is -exp(-2*x) but never solves for the constants C1 and C2 using the initial conditions. It simply verifies that the proposed answer satisfies the ODE and initial conditions, skipping the actual derivation step implied by lines 3 and 4.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution claims the answer is -exp(-2*x) without solving for the constants C1 and C2. The actual solution satisfying y(0)=-1 and y'(0)=2 is y = (1/5)e^(3x) - (6/5)e^(-2x). The provided function fails to satisfy the initial condition y'(0)=2 (it gives y'(0)=4).
gpt-oss:20b: pass 2026-09-29

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 2026-09-29 with SymPy 1.14.0.