Order and linearity of differential equations
Problem 6.420 · easy
Classify \( \displaystyle - y^{2} + y' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = \frac{1}{3 - x} \) 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.
- \[ \frac{d}{d x} \frac{1}{3 - x} - \frac{1}{\left(3 - x\right)^{2}} = 0 \]Substituting y = 1/(3 - 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to verify that y = 1/(3-x) is a solution, as the provided equation checks y' - y^2 = 0 instead of the given ODE -y^2 + y' = 0 (though they are equivalent, the verification step should explicitly match the problem statement). More critically, the problem asks to classify the ODE AND verify the solution, but the stated answer only provides the classification, omitting the verification result entirely.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to verify that y = 1/(3-x) is a solution, as the provided equation checks y' - y^2 = 0 instead of the given ODE -y^2 + y' = 0 (though they are equivalent, the verification step should explicitly match the problem statement). More critically, the problem asks to classify the ODE AND verify the solution, but the stated answer only provides the classification, omitting the verification result entirely.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to verify that y = 1/(3-x) is a solution, as the verification step is marked as an equation but the text description is misleading and the actual check is not explicitly shown as a logical step in the text. More importantly, the problem asks to classify the ODE AND verify the solution. The stated answer only gives the classification. The solution text claims substitution makes the left side 0, but does not show the work or explicitly state 'Thus, it is a solution'. However, the biggest issue is that the 'Stated answer' provided in the prompt is incomplete (missing the verification part), but the solution text *does* attempt the verification. Let's look closer. The solution says 'Substituting ... makes the left side 0'. This is a sentence describing a calculation. The equation line is marked 'proved'. The classification is correct. The verification is claimed. Is there an error? The ODE is -y^2 + y' = 0 => y' = y^2. y = 1/(3-x) => y' = 1/(3-x)^2 = y^2. So -y^2 + y' = 0. The verification is correct. The classification is correct. Why fail? The prompt asks to 'Review it'. The solution is actually correct. Wait, look at line 2: '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 technically true (A or B is true if A is true), but it's sloppy. Is it an error? No. Is it misleading? Maybe. But usually 'or' is inclusive. Let's look for a harder error. The problem asks to classify AND verify. The solution does both. The stated answer in the prompt is just 'order 1, nonlinear'. This is a mismatch between the problem request and the stated answer format, but the solution text covers both. I will mark it as pass because the math is correct and the reasoning is sound, despite the slightly loose language in line 2.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to verify that y = 1/(3-x) is a solution, as required by the problem statement. It only provides an unchecked equation without confirming the result.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.