Undetermined coefficients
Problem 6.322 · hard
Find the general solution of \( \displaystyle y'' + 4y' + 3y = 2 e^{4 x} \) 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(4*x) (the exponent is not a root, so no extra factor of x is needed).Reviewed
- \[ \frac{6 e^{4 x}}{35} + 4 \frac{d}{d x} \frac{2 e^{4 x}}{35} + \frac{d^{2}}{d x^{2}} \frac{2 e^{4 x}}{35} = 2 e^{4 x} \]Matching coefficients gives y_p = 2*exp(4*x)/35; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{- 3 x} + C_{2} e^{- x} + \frac{2 e^{4 x}}{35} \)
✓ 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 confirms the particular 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 particular solution form. The algebraic verification confirms the particular 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 applies the method of undetermined coefficients with the correct 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.