Undetermined coefficients
Problem 6.245 · hard
Find the general solution of \( \displaystyle y'' + y' - 12y = 5 e^{4 x} \) by the method of undetermined coefficients.
- \[ r^{2} + r - 12 = \left(r - 3\right) \left(r + 4\right) \]The characteristic equation has roots -4 and 3.✓ Proved
- So y_h = C₁e^(-4x) + C₂e^(3x). Guess y_p = A*exp(4*x) (the exponent is not a root, so no extra factor of x is needed).Reviewed
- \[ - \frac{15 e^{4 x}}{2} + \frac{d}{d x} \frac{5 e^{4 x}}{8} + \frac{d^{2}}{d x^{2}} \frac{5 e^{4 x}}{8} = 5 e^{4 x} \]Matching coefficients gives y_p = 5*exp(4*x)/8; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{- 4 x} + C_{2} e^{3 x} + \frac{5 e^{4 x}}{8} \)
✓ 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 particular solution form. The algebraic verification in step 3 confirms the particular solution satisfies the differential equation, and the final general solution is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and the particular solution form. The algebraic verification in step 3 confirms the particular solution satisfies the differential equation, and the final general solution is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and the particular solution form. The algebraic verification in step 3 confirms the particular solution satisfies the differential equation, and the final general solution is correct.gpt-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.