Order and linearity of differential equations
Problem 6.525 · easy
Classify \( \displaystyle 3 y + y' + 9 = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = -3 + e^{- 3 x} \) is a solution.
- The highest derivative is y′, so the order is 1.
- It is linear: y and its derivatives appear only to the first power, multiplied by functions of x. The term without y is nonzero: nonhomogeneous.
- \[ \frac{d}{d x} \left(-3 + e^{- 3 x}\right) + 3 e^{- 3 x} = 0 \]Substituting y = -3 + exp(-3*x) makes the left side 0.✓ Proved
Answer \( \text{order 1, linear and nonhomogeneous} \)
Lines: 1 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| 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 | order from sympy.ode_order; linearity from the equation's second partials in y, y′, y″; the solution checked in floating point |
Reviewers
gpt-oss:20b: fail (error) — The verification step only checks y' + 3e^{-3x}=0, omitting the 3y+9 terms of the differential equation. It does not confirm that y = -3 + e^{-3x} satisfies the full equation 3y + y' + 9 = 0.qwen3.6:27b-mlx: pass — The classification is correct: the equation is first-order, linear, and nonhomogeneous. The verification step correctly shows that the proposed solution satisfies the differential equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The classification is correct: the equation is first-order, linear, and nonhomogeneous. The verification step correctly shows that the proposed solution satisfies the differential equation.gpt-oss:20b: fail (error) 2026-10-18 — The verification step only checks y' + 3e^{-3x}=0, omitting the 3y+9 terms of the differential equation. It does not confirm that y = -3 + e^{-3x} satisfies the full equation 3y + y' + 9 = 0.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The classification is incorrect. The equation 3y + y' + 9 = 0 is nonhomogeneous because of the constant term 9, but the solution verification fails:gpt-oss:20b: fail (error) 2026-10-15 — The verification step substitutes only y′ + 3e^{−3x} = 0, omitting the 3y + 9 terms from the original differential equation. Thus it does not correctly check that y = −3 + e^{−3x} satisfies 3y + y′ + 9 = 0.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The verification step is algebraically incorrect. Substituting y = -3 + e^{-3x} into 3y + y' + 9 yields 3(-3 + e^{-3x}) + (-3e^{-3x}) + 9 = 0, but the solution's equation shows 3*exp(-3*x) instead of the correct derivative term -3*exp(-3*x), implying a sign error in the derivative calculation.gpt-oss:20b: fail (misleading) 2026-10-14 — The verification step omits the constant term +9, so the student would incorrectly think the proposed solution satisfies the equation without checking the full left‑hand side. The correct substitution yields y' + 3y + 9 = 0, which indeed equals zero for y = -3 + e^{-3x}.
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/de_classify, checked 2026-10-18 with SymPy 1.14.0.