Order and linearity of differential equations
Problem 6.563 · easy
Classify \( \displaystyle 2 x + y + y'' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = - 2 x + \sin{\left(x \right)} \) is a solution.
- The highest derivative is y′′, so the order is 2.
- 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.
- \[ \sin{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(- 2 x + \sin{\left(x \right)}\right) = 0 \]Substituting y = -2*x + sin(x) makes the left side 0.✓ Proved
Answer \( \text{order 2, linear and nonhomogeneous} \)
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 2x term and the y term in the equation, so it does not correctly show that y = -2x + sin(x) satisfies 2x + y + y'' = 0.qwen3.6:27b-mlx: pass — The classification of the differential equation is correct (order 2, linear, nonhomogeneous). The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The classification of the differential equation is correct (order 2, linear, nonhomogeneous). The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the equation.gpt-oss:20b: fail (error) 2026-10-18 — The verification step omits the 2x term and the y term in the equation, so it does not correctly show that y = -2x + sin(x) satisfies 2x + y + y'' = 0.qwen3.6:27b-mlx: pass 2026-10-16 — The classification of the differential equation is correct, and the verification step correctly shows that the proposed function satisfies the equation.gpt-oss:20b: fail (error) 2026-10-16 — The verification step omits the 2x term; substituting y = -2x + sin(x) into 2x + y + y'' gives 2x + (-2x + sin x) + (-sin x) = 0, so the calculation is incomplete and incorrect.qwen3.6:27b-mlx: pass 2026-10-15 — The classification of the differential equation is correct, and the verification step correctly identifies that the proposed function satisfies the equation.gpt-oss:20b: fail (error) 2026-10-15 — The verification step omits the terms 2x and y, so it does not correctly check that y = -2x + sin(x) satisfies the differential equation.
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.