Home›Calculus 2›Order and linearity of differential equations›Problem 6.412

Order and linearity of differential equations

Problem 6.412 · easy

Classify \( \displaystyle - y^{2} + y' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = \frac{1}{2 - x} \) is a solution.

The highest derivative is y′, so the order is 1.
It is nonlinear: y appears squared or multiplied by its own derivative.
\[ \frac{d}{d x} \frac{1}{2 - x} - \frac{1}{\left(2 - x\right)^{2}} = 0 \]
Substituting y = 1/(2 - x) makes the left side 0.✓ Proved

Answer \( \text{order 1, nonlinear} \)

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 solution fails to verify that y = 1/(2-x) is a solution, as required by the problem statement. It only classifies the ODE.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to verify that y = 1/(2-x) is a solution, as required by the problem statement. It only classifies the ODE.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution fails to verify that y = 1/(2-x) is a solution, as the provided equation checks y' - y^2 = 0 instead of the problem's -y^2 + y' = 0 (though algebraically equivalent, the verification step is incomplete without explicitly showing the substitution into the original form or acknowledging the equivalence). More critically, the problem asks to classify the ODE, but the 'Stated answer' only provides the classification, while the solution steps do not explicitly conclude the classification in a final statement, and step 2's reasoning 'y appears squared or multiplied by its own derivative' is slightly imprecise (it is y squared, not multiplied by derivative). However, the main issue is that the solution does not explicitly state the final classification as requested by the problem prompt in a concluding sentence, and the verification equation is technically for a rearranged form. Let's look closer. The problem asks to classify AND verify. The solution has steps for order, linearity, and a verification equation. Step 2 says 'It is nonlinear'. This is a sentence. Step 3 is an equation. The stated answer is 'order 1, nonlinear'. The solution misses the 'homogeneous' part of the classification request (though for nonlinear, homogeneous is often not defined or trivial, but the prompt asks 'if so, whether it is homogeneous'). Since it is nonlinear, the 'if so' condition fails, so we don't need to check homogeneity. But the solution doesn't explicitly say 'It is not linear, so the question of homogeneity does not apply' or similar. It just stops. Also, the verification equation `Derivative(1/(2 - x), x) - 1/(2 - x)**2 = 0` corresponds to `y' - y^2 = 0`. The problem is `-y^2 + y' = 0`. These are the same. So the verification is correct. The classification is correct. Is there an error? Step 2 says 'y appears squared or multiplied by its own derivative'. In `-y^2 + y' = 0`, y is squared. It is NOT multiplied by its own derivative. The 'or' makes the statement true, but it's a bit sloppy. However, the bigger issue is likely the missing explicit conclusion about the 'homogeneous' part of the prompt, or simply that the solution is incomplete. Let's look at the 'Stated answer'. It is just 'order 1, nonlinear'. The problem asks to classify order, linearity, and homogeneity. The solution identifies order 1 and nonlinear. It does not address homogeneity. For a nonlinear ODE, the term 'homogeneous' is ambiguous (could mean homogeneous in y and y', or homogeneous function). Usually, in introductory calculus, 'homogeneous' refers to linear homogeneous. Since it's nonlinear, it's not linear homogeneous. The solution should probably state that. But is it an error? The prompt asks 'if so [linear], whether it is homogeneous'. Since it is not linear, we don't need to answer the second part. So the classification is complete. The verification is correct. Why would it fail? Maybe the verification equation is marked 'proved' but the text doesn't explicitly say 'Thus, y=... is a solution'. Step 3 is an equation. The sentence 'Substituting y = 1/(2 - x) makes the left side 0' is part of the equation line? No, the format is `[equation, proved] ... -- Comment`. The comment is not a sentence in the list. The solution lacks a final sentence concluding the verification. But usually, if the equation is proved, it's fine. Let's look at Step 2 again. 'It is nonlinear: y appears squared or multiplied by its own derivative.' This is a valid reason. Is there a subtle error? The problem is `-y^2 + y' = 0`. This can be written as `y' = y^2`. This is a separable equation. The classification is correct. I will mark it as pass unless I find a specific error. Wait, the prompt says 'Classify ... its order, whether it is linear and, if so, whether it is homogeneous.' The solution says 'order 1' and 'nonlinear'. It does not explicitly say 'not homogeneous' or 'homogeneity not applicable'. But the 'if so' clause handles this. I'll give it a pass. Actually, looking at similar tasks, often 'style' or 'misleading' is used for incomplete explanations. But 'error' is for wrong math. The math is right. The logic is right. I will pass it.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the ODE is nonlinear, but the equation y' - y^2 = 0 is actually nonlinear only because of the y^2 term; however, the standard de
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.