Undetermined coefficients
Problem 6.324 · hard
Solve \( \displaystyle y'' + y' - 6y = -10 \) with \( \displaystyle y(0) = -3 \), \( \displaystyle y'(0) = 1 \).
- \[ r^{2} + r - 6 = \left(r - 2\right) \left(r + 3\right) \]The characteristic equation has roots -3 and 2.✓ Proved
- So y_h = C₁e^(-3x) + C₂e^(2x). Guess y_p = A.Reviewed
- \[ \frac{d}{d x} \frac{5}{3} + \frac{d^{2}}{d x^{2}} \frac{5}{3} - 10 = -10 \]Matching coefficients gives y_p = 5/3; it satisfies the equation.✓ Proved
- \[ \left[\begin{matrix}\left. - \frac{13 e^{2 x}}{5} + \frac{5}{3} - \frac{31 e^{- 3 x}}{15} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- \frac{13 e^{2 x}}{5} + \frac{5}{3} - \frac{31 e^{- 3 x}}{15}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}-3\\1\end{matrix}\right] \]The initial conditions fix C₁ = -31/15 and C₂ = -13/5.✓ Proved
Answer \( y = - \frac{13 e^{2 x}}{5} + \frac{5}{3} - \frac{31 e^{- 3 x}}{15} \)
✓ Nihil obstat Lines: 3 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 |
| 4 | ✓ 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 | SymPy's dsolve with the same initial conditions agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the homogeneous and particular solutions, and the constants are determined correctly to satisfy the initial conditions.
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 and particular solutions, and the constants are determined correctly to satisfy the initial conditions.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the homogeneous and particular solutions, and the constants are determined correctly to satisfy the initial conditions.
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.