Second-order initial and boundary value problems
Problem 6.808 · medium
Solve \( \displaystyle 9 y - 6 y' + y'' = 0 \) with \( \displaystyle y(0) = 4,\ y(1) = -4 \).
- \[ m^{2} - 6 m + 9 = \left(m - 3\right)^{2} \]The characteristic equation has a repeated root 3.✓ Proved
- So y = C₁exp(3*x) + C₂x*exp(3*x); the two conditions fix C₁ and C₂.
- \[ 9 \left(- 4 x e^{3} - 4 x + 4 e^{3}\right) e^{3 x - 3} - 6 \frac{d}{d x} \left(- 4 x e^{3} - 4 x + 4 e^{3}\right) e^{3 x - 3} + \frac{d^{2}}{d x^{2}} \left(- 4 x e^{3} - 4 x + 4 e^{3}\right) e^{3 x - 3} = 0 \]It satisfies the equation ...✓ Proved
- \[ \left. 4 \left(- x e^{3} - x + e^{3}\right) e^{3 x - 3} \right|_{\substack{ x=0 }} = 4 \]... y(0) = 4 ...✓ Proved
- \[ \left. 4 \left(- x e^{3} - x + e^{3}\right) e^{3 x - 3} \right|_{\substack{ x=1 }} = -4 \]y(1) = -4.✓ Proved
Answer \( y = 4 \left(- x e^{3} - x + e^{3}\right) e^{3 x - 3} \)
Lines: 4 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. How we verify
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 |
| 5 | ✓ 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 | finite differences of the answer satisfy the equation, and the conditions hold in floating point |
Reviewers
gpt-oss:20b: fail (error) — The proposed closed‑form solution y=4(-x e^3 - x + e^3)e^{3x-3} is not of the required form C1e^{3x}+C2xe^{3x}; it contains an extra x‑dependent factor multiplying e^{3x}. Thus the constants C1 and C2 cannot be determined from the given boundary conditions, and the solution does not satisfy the differential equation for all x.qwen3.6:27b-mlx: pass — The solution correctly identifies the repeated root of the characteristic equation, states the correct general form of the solution, and verifies that the specific answer satisfies both the differential equation and the boundary conditions.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the repeated root of the characteristic equation, states the correct general form of the solution, and verifies that the specific answer satisfies both the differential equation and the boundary conditions.gpt-oss:20b: fail (error) 2026-11-07 — The proposed closed‑form solution y=4(-x e^3 - x + e^3)e^{3x-3} is not of the required form C1e^{3x}+C2xe^{3x}; it contains an extra x‑dependent factor multiplying e^{3x}. Thus the constants C1 and C2 cannot be determined from the given boundary conditions, and the solution does not satisfy the differential equation for all x.qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the repeated root of the characteristic equation, states the general form of the solution, and verifies that the specific answer satisfies both the differential equation and the boundary conditions.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the repeated root of the characteristic equation, sets up the general solution, and verifies that the stated answer satisfies both the differential equation and the boundary conditions.gpt-oss:20b: pass 2026-11-06
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/second_order_ivp_bvp, checked 2026-11-07 with SymPy 1.14.0.