Second-order linear equations
Problem 6.107 · hard
Solve \( \displaystyle y'' - 4y' + 5y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = 0 \).
- 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.
- \[ \left(- 30 \sin{\left(x \right)} + 15 \cos{\left(x \right)}\right) e^{2 x} - 4 \frac{d}{d x} \left(- 6 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{2 x} + \frac{d^{2}}{d x^{2}} \left(- 6 \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{2 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ 3 \]y(0).
- \[ 0 \]y'(0).
Answer \( y = 3 \left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} \)
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: fail (error) — The proposed solution uses the wrong constant factor: it claims the solution is 15 e^{2x}(-2 sin x+cos x) instead of the correct 3 e^{2x}(-2 sin x+cos x). Consequently the initial‑value checks are inconsistent with the stated solution.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 a specific solution without showing the calculation (C1=3, C2=-6), making the derivation incomplete and potentially confusing for a student.
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 a specific solution without showing the calculation (C1=3, C2=-6), making the derivation incomplete and potentially confusing for a student.gpt-oss:20b: fail (error) 2026-09-27 — The proposed solution uses the wrong constant factor: it claims the solution is 15 e^{2x}(-2 sin x+cos x) instead of the correct 3 e^{2x}(-2 sin x+cos x). Consequently the initial‑value checks are inconsistent with the stated solution.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the general solution formula for complex conjugate roots, and verifies the initial conditions and differential equation.gpt-oss:20b: fail (error) 2026-09-26 — The proposed solution 15*(-2*sin(x)+cos(x))*exp(2*x) does not satisfy the initial conditions y(0)=3 and y'(0)=0. The correct constant factor should be 3, not 15. The subsequent verification steps are inconsistent with the stated solution.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the characteristic roots, applies the general form for complex roots, and determines the constants satisfying the initial conditions. The final answer matches the stated answer.gpt-oss:20b: fail (error) 2026-09-26 — The proposed particular solution contains an incorrect constant factor (15 instead of 3), leading to wrong initial‑value verification. The subsequent statements about y(0) and y'(0) are also inconsistent with the correct solution.
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.