Undetermined coefficients
Problem 6.410 · hard
Find the general solution of \( \displaystyle y'' - y' - 6y = 12 \) by the method of undetermined coefficients.
- \[ r^{2} - r - 6 = \left(r - 3\right) \left(r + 2\right) \]The characteristic equation has roots 3 and -2.✓ Proved
- So y_h = C₁e^(3x) + C₂e^(-2x). Guess y_p = A.Reviewed
- \[ - \frac{d}{d x} \left(-2\right) + \frac{d^{2}}{d x^{2}} \left(-2\right) + 12 = 12 \]Matching coefficients gives y_p = -2; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{3 x} + C_{2} e^{- 2 x} - 2 \)
Lines: 2 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | differs from SymPy's dsolve only by a solution of the homogeneous equation |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the homogeneous solution and the particular solution. The logic is sound, although the notation in step 3 is non-standard, the result is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the homogeneous solution and the particular solution. The logic is sound, although the notation in step 3 is non-standard, the result is correct.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (misleading) 2026-10-13 — The solution skips the crucial step of substituting the guess y_p = A into the differential equation to solve for A. It asserts the result without showing the work, which fails to demonstrate the method of undetermined coefficients.
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/undetermined_coefficients, checked 2026-10-13 with SymPy 1.14.0.