Order and linearity of differential equations
Problem 6.736 · easy
Classify \( \displaystyle - 2 y + y' - 2 = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = e^{2 x} - 1 \) 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.
- \[ - 2 e^{2 x} + \frac{d}{d x} \left(e^{2 x} - 1\right) = 0 \]Substituting y = exp(2*x) - 1 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 omits the constant term –2 in the differential equation, so the substituted expression is not the correct left‑hand side of the original equation.qwen3.6:27b-mlx: pass — The classification is correct: the equation is first-order, linear, and nonhomogeneous. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the identity.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25 — The classification is correct: the equation is first-order, linear, and nonhomogeneous. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the identity.gpt-oss:20b: fail (error) 2026-10-25 — The verification step omits the constant term –2 in the differential equation, so the substituted expression is not the correct left‑hand side of the original equation.qwen3.6:27b-mlx: pass 2026-10-24 — The classification is correct: the equation is first-order, linear, and non-homogeneous. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the identity.gpt-oss:20b: inconclusive 2026-10-24 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The verification step omits the constant terms from the original differential equation. Substituting \(y=e^{2x}-1\) into \(-2y+y'-2=0\) yields \(-2(e^{2x}qwen3.6:27b-mlx: pass 2026-10-24 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation to show it satisfies the identity.gpt-oss:20b: pass 2026-10-24
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 2026-10-25 with SymPy 1.14.0.