Undetermined coefficients
Problem 6.242 · hard
Find the general solution of \( \displaystyle y'' + 3y' + 2y = 2 \cos{\left(2 x \right)} \) by the method of undetermined coefficients.
- \[ r^{2} + 3 r + 2 = \left(r + 1\right) \left(r + 2\right) \]The characteristic equation has roots -1 and -2.✓ Proved
- So y_h = C₁e^(-1x) + C₂e^(-2x). Guess y_p = A*cos(2*x) + B*sin(2*x).Reviewed
- \[ \frac{3 \sin{\left(2 x \right)}}{5} - \frac{\cos{\left(2 x \right)}}{5} + 3 \frac{d}{d x} \left(\frac{3 \sin{\left(2 x \right)}}{10} - \frac{\cos{\left(2 x \right)}}{10}\right) + \frac{d^{2}}{d x^{2}} \left(\frac{3 \sin{\left(2 x \right)}}{10} - \frac{\cos{\left(2 x \right)}}{10}\right) = 2 \cos{\left(2 x \right)} \]Matching coefficients gives y_p = 3*sin(2*x)/10 - cos(2*x)/10; it satisfies the equation.✓ Proved
Answer \( y = C_{1} e^{- x} + C_{2} e^{- 2 x} + \frac{3 \sin{\left(2 x \right)}}{10} - \frac{\cos{\left(2 x \right)}}{10} \)
✓ 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 form of the particular solution. The coefficients for the particular solution are correct, and the final general solution matches the stated answer.
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 for the particular solution are correct, and the final general solution matches the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the homogeneous solution and the form of the particular solution. The final result matches the stated answer, and the logic is sound.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.