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).✓ Proved
- \[ -3 \]y'(0).✓ Proved
Answer \( y = \left(3 x + 2\right) e^{- 3 x} \)
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: fail (misleading) — 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 — 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.
Every verdict on record (4)
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 2026-09-29 with SymPy 1.14.0.