Second-order linear equations
Problem 6.155 · hard
Solve \( \displaystyle y'' + 4y' + 5y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = -1 \).
- Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
- \[ \left(m + 2 - i\right) \left(m + 2 + i\right) = m^{2} + 4 m + 5 \]Roots: -2 - I, -2 + I.✓ Proved
- Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
- Use y(0) and y'(0) to find C₁ and C₂.
- \[ \left(25 \sin{\left(x \right)} + 15 \cos{\left(x \right)}\right) e^{- 2 x} + 4 \frac{d}{d x} \left(5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- 2 x} + \frac{d^{2}}{d x^{2}} \left(5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- 2 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ 3 \]y(0).
- \[ -1 \]y'(0).
Answer \( y = \left(5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- 2 x} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions, jumping directly to the final answer. While the final answer is verified to satisfy the ODE and ICs, the logical step of solving the system for C1 and C2 is missing, making the derivation incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly derive the constants C1 and C2 from the initial conditions, jumping directly to the final answer. While the final answer is verified to satisfy the ODE and ICs, the logical step of solving the system for C1 and C2 is missing, making the derivation incomplete.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the characteristic roots, applies the general solution formula for complex roots, and verifies that the specific constants satisfy both the differential equation and the initial conditions.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The proposed solution multiplies the correct expression by an extra factor of 5, yielding y(x)=5e^{-2x}(5\sin x+3\cos x) instead of the correct y(x)=e^{-2qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the characteristic equation, roots, and general form. The constants are determined correctly to satisfy the initial conditions, and the final answer matches the stated answer.gpt-oss:20b: fail (error) 2026-09-28 — The proposed solution in step 5 is incorrectly multiplied by an extra factor of 5, so the expression shown does not satisfy the differential equation. The correct solution is (5 sin x + 3 cos x) e^(−2x).
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.