Home›Calculus 2›Order and linearity of differential equations›Problem 6.565

Order and linearity of differential equations

Problem 6.565 · easy

Classify \( \displaystyle 3 y - 4 y' + y'' = 0 \): 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.
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.
\[ 3 e^{3 x} - 4 \frac{d}{d x} e^{3 x} + \frac{d^{2}}{d x^{2}} e^{3 x} = 0 \]
Substituting y = exp(3*x) makes the left side 0.✓ Proved

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

Lines: 1 proved, 2 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: pass
qwen3.6:27b-mlx: fail (error) — The verification step is incorrect: substituting y = e^{3x} into 3y - 4y' + y'' yields 3e^{3x} - 12e^{3x} + 9e^{3x} = 0, which is true, but the solution text claims the substitution makes the left side 0 without showing the calculation, and more importantly, the problem asks to verify it is a solution. Wait, 3(1) - 4(3) + 9 = 3 - 12 + 9 = 0. It IS a solution. Let me re-read the prompt. 'Verify that y = e^{3x} is a solution.' The equation is 3y - 4y' + y'' = 0. Characteristic equation: r^2 - 4r + 3 = 0 => (r-3)(r-1)=0. So r=3 is a root. y=e^{3x} is indeed a solution. The classification is correct. The verification is correct. Why did I think it was wrong? Let me check the algebra again. y=e^{3x}, y'=3e^{3x}, y''=9e^{3x}. 3(e^{3x}) - 4(3e^{3x}) + 9e^{3x} = (3 - 12 + 9)e^{3x} = 0. It works. The solution is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The verification step is incorrect: substituting y = e^{3x} into 3y - 4y' + y'' yields 3e^{3x} - 12e^{3x} + 9e^{3x} = 0, which is true, but the solution text claims the substitution makes the left side 0 without showing the calculation, and more importantly, the problem asks to verify it is a solution. Wait, 3(1) - 4(3) + 9 = 3 - 12 + 9 = 0. It IS a solution. Let me re-read the prompt. 'Verify that y = e^{3x} is a solution.' The equation is 3y - 4y' + y'' = 0. Characteristic equation: r^2 - 4r + 3 = 0 => (r-3)(r-1)=0. So r=3 is a root. y=e^{3x} is indeed a solution. The classification is correct. The verification is correct. Why did I think it was wrong? Let me check the algebra again. y=e^{3x}, y'=3e^{3x}, y''=9e^{3x}. 3(e^{3x}) - 4(3e^{3x}) + 9e^{3x} = (3 - 12 + 9)e^{3x} = 0. It works. The solution is sound.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The classification is incorrect: the equation is not homogeneous because the coefficient of y is 3 (non-zero), meaning the term 3y is present. A lin
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The classification is correct, but the verification step is mathematically false. Substituting y = e^{3x} into 3y - 4y' + y'' yields 3e^{3x} - 12e^{
gpt-oss:20b: pass 2026-10-15

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.