Home›Calculus 2›Order and linearity of differential equations›Problem 6.411

Order and linearity of differential equations

Problem 6.411 · easy

Classify \( \displaystyle - 4 y + y'' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = e^{- 2 x} \) 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
\[ \frac{d^{2}}{d x^{2}} e^{- 2 x} - 4 e^{- 2 x} = 0 \]
Substituting y = exp(-2*x) makes the left side 0.✓ Proved

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

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The classification is correct and the verification step is algebraically sound.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The classification is correct and the verification step correctly substitutes the proposed solution into the differential equation to show it satisfies the identity.
gpt-oss:20b: pass 2026-10-14

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.