Undetermined coefficients
Problem 6.249 · hard
Solve \( \displaystyle y'' - 5y' + 6y = 12 \) with \( \displaystyle y(0) = 2 \), \( \displaystyle y'(0) = -2 \).
- \[ r^{2} - 5 r + 6 = \left(r - 3\right) \left(r - 2\right) \]The characteristic equation has roots 3 and 2.✓ Proved
- So y_h = C₁e^(3x) + C₂e^(2x). Guess y_p = A.
- \[ - 5 \frac{d}{d x} 2 + \frac{d^{2}}{d x^{2}} 2 + 12 = 12 \]Matching coefficients gives y_p = 2; it satisfies the equation.✓ Proved
- \[ \left[\begin{matrix}\left. - 2 e^{3 x} + 2 e^{2 x} + 2 \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- 2 e^{3 x} + 2 e^{2 x} + 2\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}2\\-2\end{matrix}\right] \]The initial conditions fix C₁ = -2 and C₂ = 2.✓ Proved
Answer \( y = - 2 e^{3 x} + 2 e^{2 x} + 2 \)
Lines: 3 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 |
| 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: fail (error) — The solution jumps from the general form to the final answer without showing the derivation of the constants C1 and C2. Step 4 verifies the final answer but does not demonstrate the system of equations used to find C1 = -2 and C2 = 2, leaving the student without the method for solving for constants.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution jumps from the general form to the final answer without showing the derivation of the constants C1 and C2. Step 4 verifies the final answer but does not demonstrate the system of equations used to find C1 = -2 and C2 = 2, leaving the student without the method for solving for constants.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution presents the final answer in step 4 without showing the derivation of the constants C1 and C2 from the initial conditions. Specifically, it skips the system of equations C1 + C2 = 2 and 3C1 + 2C2 = -2, jumping directly to the result. While the final answer is correct, the logical gap makes it impossible to verify the method used to find the constants, and it fails to demonstrate the standard procedure for solving IVPs.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.