Home›Calculus 3›Second-order linear equations›Problem 6.41

Second-order linear equations

Problem 6.41 · medium

Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 2 \).

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

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

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 actually solve for the constants C1 and C2. It jumps from the general form to the final answer without showing the algebraic steps (y(0)=1 => C1=1; y'(0)=2 => C2=-1). While the final answer is correct, the derivation is missing, making it an incomplete solution.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to actually solve for the constants C1 and C2. It jumps from the general form to the final answer without showing the algebraic steps (y(0)=1 => C1=1; y'(0)=2 => C2=-1). While the final answer is correct, the derivation is missing, making it an incomplete solution.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to actually solve for the constants C1 and C2. It jumps from the general form to the final answer without showing the calculation (C1=1, C2=-3), making the derivation incomplete and logically disconnected.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, applies the standard form for the general solution, and verifies the initial conditions and differential equation. The steps are logically sound and mathematically correct.
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 2026-10-13 with SymPy 1.14.0.