Home›Calculus 3›Second-order linear equations›Problem 6.32

Second-order linear equations

Problem 6.32 · hard

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

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.
\[ \left(36 x - 18\right) e^{3 x} - 6 \frac{d}{d x} \left(4 x - 2\right) e^{3 x} + \frac{d^{2}}{d x^{2}} \left(4 x - 2\right) e^{3 x} = 0 \]
The solution satisfies the equation.✓ Proved
\[ -2 \]
y(0).
\[ -2 \]
y'(0).

Answer \( y = \left(4 x - 2\right) e^{3 x} \)

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 fails to explicitly solve for the constants C1 and C2. It jumps from the general form to the specific answer without showing the system of equations derived from the initial conditions, making the derivation incomplete.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly solve for the constants C1 and C2. It jumps from the general form to the specific answer without showing the system of equations derived from the initial conditions, making the derivation incomplete.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions, jumping directly to the final answer without showing the system of equations. While the final answer is correct, the logical gap between step 4 and the verification steps makes the derivation incomplete and potentially confusing for a student learning the method.
gpt-oss:20b: pass 2026-09-27
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. While the final answer is verified to satisfy the ODE and ICs, the logical step of solving the system for C1 and C2 is missing, making the derivation incomplete.
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; it merely states that the proposed function satisfies the differential equation and lists the initial values without showing how they were obtained. This omission could mislead a student into thinking the constants are arbitrary or that the verification alone suffices.

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.