Second-order linear equations
Problem 6.138 · hard
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.Reviewed
- \[ \left(m + 3\right)^{2} = m^{2} + 6 m + 9 \]Roots: -3.✓ Proved
- A repeated root r gives y = (C₁ + C₂x)e^(rx).Reviewed
- Use y(0) and y'(0) to find C₁ and C₂.Reviewed
- \[ \left(- 45 x - 9\right) e^{- 3 x} + 6 \frac{d}{d x} \left(- 5 x - 1\right) e^{- 3 x} + \frac{d^{2}}{d x^{2}} \left(- 5 x - 1\right) e^{- 3 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ -1 \]y(0).✓ Proved
- \[ -2 \]y'(0).✓ Proved
Answer \( y = \left(- 5 x - 1\right) e^{- 3 x} \)
✓ Nihil obstat Lines: 4 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the repeated root, states the general form, and verifies that the specific solution satisfies both the differential equation and the initial conditions.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the repeated root, states the general form, and verifies that the specific solution satisfies both the differential equation and the initial conditions.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the repeated root of the characteristic equation, applies the standard form for repeated roots, and verifies that the final answer satisfies both the differential equation and the initial conditions.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.