Undetermined coefficients
Problem 6.283 · hard
Find the general solution of \( \displaystyle y'' + 4y' + 3y = 3 e^{- 2 x} \) by the method of undetermined coefficients.
- \[ r^{2} + 4 r + 3 = \left(r + 1\right) \left(r + 3\right) \]The characteristic equation has roots -3 and -1.✓ Proved
- So y_h = C₁e^(-3x) + C₂e^(-1x). Guess y_p = A*exp(-2*x) (the exponent is not a root, so no extra factor of x is needed).Reviewed
- \[ 4 \frac{d}{d x} \left(- 3 e^{- 2 x}\right) + \frac{d^{2}}{d x^{2}} \left(- 3 e^{- 2 x}\right) - 9 e^{- 2 x} = 3 e^{- 2 x} \]Matching coefficients gives y_p = -3*exp(-2*x); it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{- 3 x} + C_{2} e^{- x} - 3 e^{- 2 x} \)
✓ 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 verification step confirms the particular solution satisfies the differential equation, leading to the correct general solution.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the homogeneous solution and the form of the particular solution. The verification step confirms the particular solution satisfies the differential equation, leading to the correct general solution.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the homogeneous solution and the form of the particular solution. The final check confirms the particular solution satisfies the differential equation, and the general solution is the correct sum.
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 2026-10-07 with SymPy 1.14.0.