Second-order linear equations
Problem 6.80 · hard
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -3 \).
- 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.
- \[ \left(- 36 x - 12\right) e^{- 2 x} + 4 \frac{d}{d x} \left(- 9 x - 3\right) e^{- 2 x} + \frac{d^{2}}{d x^{2}} \left(- 9 x - 3\right) e^{- 2 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ -3 \]y(0).
- \[ -3 \]y'(0).
Answer \( y = 3 \left(- 3 x - 1\right) e^{- 2 x} \)
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 verification step 5 incorrectly multiplies the proposed solution by 12 before substituting into the differential equation, which would not satisfy the ODE. The correct substitution should use the solution without the extraneous factor.qwen3.6:27b-mlx: fail (error) — 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 step connecting the general solution to the specific solution is missing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — 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 step connecting the general solution to the specific solution is missing.gpt-oss:20b: fail (error) 2026-09-28 — The verification step 5 incorrectly multiplies the proposed solution by 12 before substituting into the differential equation, which would not satisfy the ODE. The correct substitution should use the solution without the extraneous factor.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 a specific solution and then verifies it, omitting the actual calculation steps required to solve for the constants.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-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.gpt-oss:20b: fail (error) 2026-09-26 — The proposed solution 12*(-3*x-1)*exp(-2*x) does not satisfy the differential equation or the initial conditions. The correct solution is 3*(-3*x-1)*exp(-2*x).
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.