Home›Calculus 2›Order and linearity of differential equations›Problem 6.564

Order and linearity of differential equations

Problem 6.564 · easy

Classify \( \displaystyle - 3 y + y' + 6 = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = e^{3 x} + 2 \) is a solution.

The highest derivative is y′, so the order is 1.
Reviewed
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.
Reviewed
\[ - 3 e^{3 x} + \frac{d}{d x} \left(e^{3 x} + 2\right) = 0 \]
Substituting y = exp(3*x) + 2 makes the left side 0.✓ Proved

Answer \( \text{order 1, linear and nonhomogeneous} \)

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 classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation to show it satisfies the identity.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the equation to show it satisfies the identity.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The classification of the differential equation is correct, and the verification step correctly substitutes the proposed solution into the original equation to show it satisfies the identity.
gpt-oss:20b: fail (error) 2026-10-16 — The verification step omits the constant term +6; the substituted expression should be -3(e^{3x}+2)+3e^{3x}+6, which indeed equals 0. The given equation is incomplete.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The verification step is mathematically incorrect. Substituting y = e^{3x} + 2 into -3y + y' + 6 yields -3(e^{3x} + 2) + 3e^{3x} + 6 = 0, which is correct. However, the equation shown in line 3 is -3*exp(3*x) + Derivative(exp(3*x) + 2, x) = 0, which simplifies to -3e^{3x} + 3e^{3x} = 0. This equation ignores the constant terms (-6 from -3y and +6 from the original ODE), effectively verifying a different equation (-3y + y' = 0) rather than the one given.
gpt-oss:20b: fail (error) 2026-10-15 — The verification step omitted the constant term +6 from the differential equation, so the substitution does not actually test the given 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.