Home›Calculus 2›Order and linearity of differential equations›Problem 6.671

Order and linearity of differential equations

Problem 6.671 · easy

Classify \( \displaystyle - 9 y + y'' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = e^{- 3 x} \) 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
\[ \frac{d^{2}}{d x^{2}} e^{- 3 x} - 9 e^{- 3 x} = 0 \]
Substituting y = exp(-3*x) 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 classification is correct and the verification step is algebraically sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The classification is correct and the verification step is algebraically sound.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The classification is correct and the verification step is algebraically sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The verification step is incorrect because the given function y = e^{-3x} is not a solution to the differential equation y'' - 9y = 0 (it satisfies y'' - 9y = 0 only if the sign is different or the exponent is different; specifically, y'' = 9e^{-3x}, so y'' - 9y = 0 is false, it equals 0? Wait. y = e^{-3x}, y' = -3e^{-3x}, y'' = 9e^{-3x}. Then y'' - 9y = 9e^{-3x} - 9e^{-3x} = 0. Actually, it IS a solution. Let me re-read the problem statement carefully. Problem: -9y + y'' = 0. This is y'' - 9y = 0. Characteristic equation r^2 - 9 = 0 => r = +/- 3. So y = e^{3x} and y = e^{-3x} are solutions. The solution provided says 'Substituting y = exp(-3*x) makes the left side 0'. The equation line says 'Derivative(exp(-3*x), (x, 2)) - 9*exp(-3*x) = 0'. This is algebraically correct. Why did I think it was wrong? Ah, I misread the sign in my head initially. Let's look closer. Is there any other error? The classification is correct. The verification is correct. Wait, looking at the JSON output requirements. If it is sound, say pass. Is it sound? Yes. But wait, usually these adversarial reviews find something. Let me check the 'unchecked' tag. The prompt says 'equation, unchecked' but also 'computer algebra system has already checked... do not re-check the algebra of a line marked proved'. Here it is marked 'unchecked'. Does that mean I should check it? The prompt says 'Each line... is either an EQUATION... (its result is shown; do not re-check...)' or a SENTENCE. Line 3 is marked [equation, unchecked]. This is ambiguous. However, the text says 'Substituting y = exp(-3*x) makes the left side 0'. This is a sentence describing the equation. The equation itself is `Derivative(...) - 9*... = 0`. This is true. So the solution is correct. Why would it fail? Maybe the prompt implies that 'unchecked' means I SHOULD check it? If I check it, it is correct. So it passes. Let me look for subtle errors. 'y and its derivatives appear only to the first power, multiplied by functions of x.' This is the definition of linear. Correct. 'The term without y is zero: homogeneous.' Correct. Order 2. Correct. Verification correct. I will mark it pass.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.