Order and linearity of differential equations
Problem 6.421 · easy
Classify \( \displaystyle 2 y + y' - 6 = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = 3 + e^{- 2 x} \) 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.
- \[ \frac{d}{d x} \left(3 + e^{- 2 x}\right) + 2 e^{- 2 x} = 0 \]Substituting y = 3 + exp(-2*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 omits the constant term –6 from the differential equation, so the calculation does not actually check that y=3+e^{-2x} satisfies 2y+y'-6=0.qwen3.6:27b-mlx: fail (error) — The verification step is incorrect. Substituting y = 3 + e^{-2x} into the original equation 2y + y' - 6 = 0 yields 2(3 + e^{-2x}) + (-2e^{-2x}) - 6 = 6 + 2e^{-2x} - 2e^{-2x} - 6 = 0. The solution claims the derivative is 2*exp(-2*x) (positive), which is incorrect (it should be -2*exp(-2*x)), and the equation shown does not match the original ODE structure (missing the 2y and -6 terms explicitly in the check).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The verification step is incorrect. Substituting y = 3 + e^{-2x} into the original equation 2y + y' - 6 = 0 yields 2(3 + e^{-2x}) + (-2e^{-2x}) - 6 = 6 + 2e^{-2x} - 2e^{-2x} - 6 = 0. The solution claims the derivative is 2*exp(-2*x) (positive), which is incorrect (it should be -2*exp(-2*x)), and the equation shown does not match the original ODE structure (missing the 2y and -6 terms explicitly in the check).gpt-oss:20b: fail (error) 2026-10-17 — The verification step omits the constant term –6 from the differential equation, so the calculation does not actually check that y=3+e^{-2x} satisfies 2y+y'-6=0.qwen3.6:27b-mlx: pass 2026-10-14 — The classification is correct: the equation is first-order, linear, and nonhomogeneous due to the constant term. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the identity.gpt-oss:20b: fail (error) 2026-10-14 — The verification step incorrectly substitutes y into the differential equation. It omits the 2y and -6 terms, so the conclusion that the left side equals 0 is unsupported.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The verification step is algebraically incorrect. Substituting y = 3 + e^{-2x} into 2y + y' - 6 yields 2(3 + e^{-2x}) + (-2e^{-2x}) - 6 = 0, but the solution line shows 2*exp(-2*x) + Derivative(...) = 0, omitting the constant terms (6 and -6) and incorrectly implying the variable terms alone sum to zero without the constants cancelling out. Specifically, the equation shown is 2e^{-2x} - 2e^{-2x} = 0, which ignores the '2*3 - 6' part of the substitution.gpt-oss:20b: fail (error) 2026-10-14 — The verification step substitutes y into the wrong expression; it omits the 2y and -6 terms, so the conclusion that the left side is 0 is incorrect.
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 2026-10-17 with SymPy 1.14.0.