Home›Calculus 3›Second-order linear equations›Problem 6.179

Second-order linear equations

Problem 6.179 · hard

Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -3 \).

Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
\[ \left(m + 3\right)^{2} = m^{2} + 6 m + 9 \]
Roots: -3.✓ Proved
A repeated root r gives y = (C₁ + C₂x)e^(rx).
Use y(0) and y'(0) to find C₁ and C₂.
\[ \left(27 x + 18\right) e^{- 3 x} + 6 \frac{d}{d x} \left(3 x + 2\right) e^{- 3 x} + \frac{d^{2}}{d x^{2}} \left(3 x + 2\right) e^{- 3 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ 2 \]
y(0).
\[ -3 \]
y'(0).

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

Lines: 2 proved, 5 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. How we verify

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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
7	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 skips the derivation of the constants C1 and C2, jumping directly to the final answer. While the final answer is correct, the steps shown do not demonstrate how the initial conditions were used to determine the specific values of the constants, making the solution incomplete and logically disconnected.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution skips the derivation of the constants C1 and C2, jumping directly to the final answer. While the final answer is correct, the steps shown do not demonstrate how the initial conditions were used to determine the specific values of the constants, making the solution incomplete and logically disconnected.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. The logic is sound and complete.
gpt-oss:20b: fail (misleading) 2026-09-29 — The solution never actually determines the constants C₁ and C₂ from the initial conditions; it merely states that they are found and then shows that the proposed function satisfies the differential equation and the initial data. A student would learn that the constants can be chosen arbitrarily, which is false.
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the repeated root, applies the standard form for the general solution, and verifies the initial conditions and differential equation.
gpt-oss:20b: fail (error) 2026-09-29 — The solution never determines the constants C₁ and C₂ from the initial conditions; it simply states that the proposed function satisfies the differential equation and lists the initial values without showing that the function actually meets them. This omission makes the solution incomplete and potentially misleading.

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 2026-10-13 with SymPy 1.14.0.