Second-order linear equations
Problem 6.38 · 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.Reviewed
- \[ \left(m - 2 - i\right) \left(m - 2 + i\right) = m^{2} - 4 m + 5 \]Roots: 2 - I, 2 + I.✓ Proved
- Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).Reviewed
- Use y(0) and y'(0) to find C₁ and C₂.Reviewed
- \[ \left(15 \sin{\left(x \right)} - 5 \cos{\left(x \right)}\right) e^{2 x} - 4 \frac{d}{d x} \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x} + \frac{d^{2}}{d x^{2}} \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x} = 0 \]The solution satisfies the equation.✓ Proved
- \[ -1 \]y(0).Reviewed
- \[ 1 \]y'(0).Reviewed
Answer \( y = \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x} \)
Lines: 2 proved, 5 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 7 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the characteristic equation, finds the complex roots, applies the standard form for complex conjugate roots, and verifies the initial conditions. The steps are logically sound and mathematically correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the standard form for complex conjugate roots, and verifies the initial conditions. The steps are logically sound and mathematically correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the standard form for complex conjugate roots, and verifies the final answer against the differential equation and initial conditions.gpt-oss:20b: fail (error) 2026-09-27 — The solution never determines the constants C₁ and C₂ from the initial conditions, and the statements in steps 6 and 7 are ambiguous (they list two identical values each). Thus the setup is incomplete and the conclusion that the given function satisfies the initial value problem is unsupported.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the characteristic equation, finds the complex roots, applies the standard form for complex conjugate 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 lists two numbers for each initial condition (e.g., "-1, -1" for y(0)), which is confusing and could mislead a student into thinking the value is ambiguous. The correct initial values are y(0) = –1 and y′(0) = 1, each a single number.
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.