Undetermined coefficients
Problem 6.243 · hard
Find the general solution of \( \displaystyle y'' - 5y' + 6y = - 4 \sin{\left(2 x \right)} \) by the method of undetermined coefficients.
- \[ r^{2} - 5 r + 6 = \left(r - 3\right) \left(r - 2\right) \]The characteristic equation has roots 2 and 3.✓ Proved
- So y_h = C₁e^(2x) + C₂e^(3x). Guess y_p = A*cos(2*x) + B*sin(2*x).
- \[ - \frac{6 \sin{\left(2 x \right)}}{13} - \frac{30 \cos{\left(2 x \right)}}{13} - 5 \frac{d}{d x} \left(- \frac{\sin{\left(2 x \right)}}{13} - \frac{5 \cos{\left(2 x \right)}}{13}\right) + \frac{d^{2}}{d x^{2}} \left(- \frac{\sin{\left(2 x \right)}}{13} - \frac{5 \cos{\left(2 x \right)}}{13}\right) = - 4 \sin{\left(2 x \right)} \]Matching coefficients gives y_p = -sin(2*x)/13 - 5*cos(2*x)/13; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{2 x} + C_{2} e^{3 x} - \frac{\sin{\left(2 x \right)}}{13} - \frac{5 \cos{\left(2 x \right)}}{13} \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The particular solution guess ignores the fact that r=2 is a root of the characteristic equation, so the trial function must be multiplied by x. The provided guess A cos(2x)+B sin(2x) is therefore incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the homogeneous solution and the form of the particular solution. The coefficients are calculated correctly, and the final general solution is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and the form of the particular solution. The coefficients are calculated correctly, and the final general solution is accurate.gpt-oss:20b: fail (error) 2026-10-05 — The particular solution guess ignores the fact that r=2 is a root of the characteristic equation, so the trial function must be multiplied by x. The provided guess A cos(2x)+B sin(2x) is therefore incorrect.qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and uses the appropriate form for the particular solution. The coefficients are calculated correctly, and the final general solution is accurate.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.