Undetermined coefficients
Problem 6.327 · hard
Find the general solution of \( \displaystyle y'' - 3y' + 2y = x - 6 \) by the method of undetermined coefficients.
- \[ r^{2} - 3 r + 2 = \left(r - 2\right) \left(r - 1\right) \]The characteristic equation has roots 2 and 1.✓ Proved
- So y_h = C₁e^(2x) + C₂e^(1x). Guess y_p = A*x + B.Reviewed
- \[ x - 3 \frac{d}{d x} \left(\frac{x}{2} - \frac{9}{4}\right) + \frac{d^{2}}{d x^{2}} \left(\frac{x}{2} - \frac{9}{4}\right) - \frac{9}{2} = x - 6 \]Matching coefficients gives y_p = x/2 - 9/4; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{2 x} + C_{2} e^{x} + \frac{x}{2} - \frac{9}{4} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 form of the particular solution. The final general solution is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the homogeneous solution and the form of the particular solution. The final general solution is correct.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the homogeneous solution and uses the appropriate form for the particular solution. The final general solution is correct.
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 2026-10-08 with SymPy 1.14.0.