Home›Calculus 2›Order and linearity of differential equations›Problem 6.413

Order and linearity of differential equations

Problem 6.413 · easy

Classify \( \displaystyle x^{2} y'' - 2 y = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = x^{2} \) is a solution.

The highest derivative is y′′, so the order is 2.
Reviewed
It is linear: y and its derivatives appear only to the first power, multiplied by functions of x. The term without y is zero: homogeneous.
Reviewed
\[ x^{2} \frac{d^{2}}{d x^{2}} x^{2} - 2 x^{2} = 0 \]
Substituting y = x**2 makes the left side 0.✓ Proved

Answer \( \text{order 2, linear and homogeneous} \)

Lines: 1 proved, 2 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the order, linearity, and homogeneity of the differential equation. The verification step is algebraically correct and clearly stated.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the order, linearity, and homogeneity of the differential equation. The verification step is algebraically correct and clearly stated.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the order, linearity, and homogeneity based on standard definitions. The verification step correctly substitutes the proposed solution into the differential equation and confirms it satisfies the equation.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution to show it satisfies the equation.
gpt-oss:20b: fail (error) 2026-10-14 — The verification step incorrectly substitutes y=x²: the second derivative of x² is 0, so the left‑hand side becomes –2x², not 0. Thus y=x² is not a solution of the given 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 2026-10-17 with SymPy 1.14.0.