Undetermined coefficients
Problem 6.247 · hard
Find the general solution of \( \displaystyle y'' + 5y' + 4y = e^{4 x} \) by the method of undetermined coefficients.
- \[ r^{2} + 5 r + 4 = \left(r + 1\right) \left(r + 4\right) \]The characteristic equation has roots -1 and -4.✓ Proved
- So y_h = C₁e^(-1x) + C₂e^(-4x). Guess y_p = A*exp(4*x) (the exponent is not a root, so no extra factor of x is needed).Reviewed
- \[ \frac{e^{4 x}}{10} + 5 \frac{d}{d x} \frac{e^{4 x}}{40} + \frac{d^{2}}{d x^{2}} \frac{e^{4 x}}{40} = e^{4 x} \]Matching coefficients gives y_p = exp(4*x)/40; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{- x} + C_{2} e^{- 4 x} + \frac{e^{4 x}}{40} \)
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 applies the method of undetermined coefficients with the correct form for the particular solution. The algebraic verification confirms the coefficient is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and applies the method of undetermined coefficients with the correct form for the particular solution. The algebraic verification confirms the coefficient is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims that matching coefficients gives y_p = exp(4*x)/40, but the algebra is incorrect. Substituting y_p = A e^{4x} into the LHS yieldgpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.