Second-order linear equations
Problem 6.109 · hard
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -2 \).
- Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
- \[ \left(m + 2\right)^{2} = m^{2} + 4 m + 4 \]Roots: -2.✓ Proved
- A repeated root r gives y = (C₁ + C₂x)e^(rx).
- Use y(0) and y'(0) to find C₁ and C₂.
- \[ \left(- 24 x - 8\right) e^{- 2 x} + 4 \frac{d}{d x} \left(- 6 x - 2\right) e^{- 2 x} + \frac{d^{2}}{d x^{2}} \left(- 6 x - 2\right) e^{- 2 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ -2 \]y(0).
- \[ -2 \]y'(0).
Answer \( y = 2 \left(- 3 x - 1\right) e^{- 2 x} \)
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: fail (error) — The solution line 5 incorrectly substitutes 8*(-3*x-1)*exp(-2*x) instead of the correct 2*(-3*x-1)*exp(-2*x), yet claims it satisfies the differential equation. This misrepresents the actual solution and would mislead a student.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions. It jumps from the general form to the final answer without showing the system of equations (C1 = -2, C2 - 2C1 = -2) that leads to the specific solution, making the logical gap between steps 3 and 5 opaque.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions. It jumps from the general form to the final answer without showing the system of equations (C1 = -2, C2 - 2C1 = -2) that leads to the specific solution, making the logical gap between steps 3 and 5 opaque.gpt-oss:20b: fail (error) 2026-09-27 — The solution line 5 incorrectly substitutes 8*(-3*x-1)*exp(-2*x) instead of the correct 2*(-3*x-1)*exp(-2*x), yet claims it satisfies the differential equation. This misrepresents the actual solution and would mislead a student.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. Although the constants C1 and C2 are not explicitly solved for in the text, the final answer is derived correctly and verified.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the repeated root, applies the standard form for repeated roots, and verifies the initial conditions and differential equation. The steps are logically sound and mathematically correct.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 and presents an incorrect intermediate expression (8*(-3*x-1)*exp(-2*x)) that does not match the claimed final solution. This would mislead a student into believing the verification step is correct without proper justification.
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.