Undetermined coefficients
Problem 6.404 · hard
Solve \( \displaystyle y'' - 2y' - 3y = - e^{- 2 x} \) with \( \displaystyle y(0) = -3 \), \( \displaystyle y'(0) = 1 \).
- \[ r^{2} - 2 r - 3 = \left(r - 3\right) \left(r + 1\right) \]The characteristic equation has roots -1 and 3.✓ Proved
- So y_h = C₁e^(-1x) + C₂e^(3x). Guess y_p = A*exp(-2*x) (the exponent is not a root, so no extra factor of x is needed).Reviewed
- \[ - 2 \frac{d}{d x} \left(- \frac{e^{- 2 x}}{5}\right) + \frac{d^{2}}{d x^{2}} \left(- \frac{e^{- 2 x}}{5}\right) + \frac{3 e^{- 2 x}}{5} = - e^{- 2 x} \]Matching coefficients gives y_p = -exp(-2*x)/5; it satisfies the equation.✓ Proved
- \[ \left[\begin{matrix}\left. - \frac{11 e^{3 x}}{20} - \frac{9 e^{- x}}{4} - \frac{e^{- 2 x}}{5} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- \frac{11 e^{3 x}}{20} - \frac{9 e^{- x}}{4} - \frac{e^{- 2 x}}{5}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}-3\\1\end{matrix}\right] \]The initial conditions fix C₁ = -9/4 and C₂ = -11/20.✓ Proved
Answer \( y = - \frac{11 e^{3 x}}{20} - \frac{9 e^{- x}}{4} - \frac{e^{- 2 x}}{5} \)
✓ 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 solution, proposes the correct form for the particular solution, verifies it, and correctly applies the initial conditions to determine the constants.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the homogeneous solution, proposes the correct form for the particular solution, verifies it, and correctly applies the initial conditions to determine the constants.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the homogeneous solution, proposes a valid particular solution form, verifies the particular solution, and correctly applies the initial conditions to determine the constants.
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 2026-10-13 with SymPy 1.14.0.