Second-order linear equations
Problem 6.33 · hard
Solve \( \displaystyle y'' - 4y' + 5y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -1 \).
- 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.
- \[ - 5 \sqrt{2} e^{2 x} \cos{\left(x + \frac{\pi}{4} \right)} - 4 \frac{d}{d x} \left(- \sqrt{2} e^{2 x} \cos{\left(x + \frac{\pi}{4} \right)}\right) + \frac{d^{2}}{d x^{2}} \left(- \sqrt{2} e^{2 x} \cos{\left(x + \frac{\pi}{4} \right)}\right) = 0 \]The solution satisfies the equation.✓ Proved
- \[ -1 \]y(0).
- \[ -1 \]y'(0).
Answer \( y = - \sqrt{2} e^{2 x} \cos{\left(x + \frac{\pi}{4} \right)} \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The proposed solution contains an incorrect constant factor: the correct solution is \(-\sqrt{2}\,e^{2x}\cos(x+\pi/4)\), not \(-5\sqrt{2}\,e^{2x}\cos(x+\pqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly solve for the constants C1 and C2, instead jumping to the final answer. While the final answer is verified to satisfy the ODE and initial conditions, the logical step of determining the constants from the general solution is missing, making the derivation incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly solve for the constants C1 and C2, instead jumping to the final answer. While the final answer is verified to satisfy the ODE and initial conditions, the logical step of determining the constants from the general solution is missing, making the derivation incomplete.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The proposed solution contains an incorrect constant factor: the correct solution is \(-\sqrt{2}\,e^{2x}\cos(x+\pi/4)\), not \(-5\sqrt{2}\,e^{2x}\cos(x+\pqwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the characteristic roots, applies the general form for complex conjugate roots, and verifies that the specific solution satisfies both the differential equation and the initial conditions.gpt-oss:20b: fail (error) 2026-09-27 — The solution line 5 incorrectly claims that “-5*sqrt(2)*exp(2*x)*cos(x + pi/4)” satisfies the differential equation, yet the correct solution is “- sqrt(2)*exp(2*x)*cos(x + pi/4)”. Additionally, the steps for determining the constants C1 and C2 are omitted, leaving the derivation incomplete.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the characteristic roots, applies the general solution formula for complex roots, and verifies that the specific answer satisfies both the differential equation and the initial conditions.gpt-oss:20b: pass 2026-09-26
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.