Order and linearity of differential equations
Problem 6.517 · easy
Classify \( \displaystyle - x + y + y'' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = x + \sin{\left(x \right)} \) 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.
- \[ \sin{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(x + \sin{\left(x \right)}\right) = 0 \]Substituting y = 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 -x term; the substituted expression should be -x + (x+sin x) + (-sin x)=0, not just sin(x)+y''=0.qwen3.6:27b-mlx: pass — The classification is correct: the equation is second-order, linear, and non-homogeneous due to the -x term. The verification step correctly substitutes the proposed solution into the differential equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The classification is correct: the equation is second-order, linear, and non-homogeneous due to the -x term. The verification step correctly substitutes the proposed solution into the differential equation.gpt-oss:20b: fail (error) 2026-10-18 — The verification step omits the -x term; the substituted expression should be -x + (x+sin x) + (-sin x)=0, not just sin(x)+y''=0.qwen3.6:27b-mlx: pass 2026-10-15 — The classification of the differential equation is correct, and the verification step correctly demonstrates that the proposed function satisfies the equation.gpt-oss:20b: fail (misleading) 2026-10-15 — The verification step omits the -x term from the differential equation, so it does not actually confirm that y=x+sin(x) satisfies -x+y+y''=0.qwen3.6:27b-mlx: pass 2026-10-14 — The classification of the differential equation is correct (order 2, linear, nonhomogeneous). The verification step correctly identifies that substituting the proposed solution satisfies the equation.gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The verification step omits the \u2013x term from the original differential equation. The correct substitution should start from \u2013x + y + y'' = 0, le
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.