Home›Calculus 2›Order and linearity of differential equations›Problem 6.566

Order and linearity of differential equations

Problem 6.566 · easy

Classify \( \displaystyle - y^{2} + y' = 0 \): its order, whether it is linear and, if so, whether it is homogeneous. Then verify that \( \displaystyle y = \frac{1}{4 - x} \) 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.
\[ \frac{d}{d x} \frac{1}{4 - x} - \frac{1}{\left(4 - x\right)^{2}} = 0 \]
Substituting y = 1/(4 - 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 claims the ODE is nonlinear, but the equation y' - y^2 = 0 is actually nonlinear due to the y^2 term, so the classification is correct. However, the reasoning in step 2 is flawed: it states 'y appears squared OR multiplied by its own derivative'. While y^2 is present, the phrase 'or multiplied by its own derivative' suggests a condition for nonlinearity that is not met here (y is not multiplied by y'), and more importantly, the verification step is incomplete. The problem asks to verify the solution, but the solution only shows the algebraic check without explicitly stating that the substitution satisfies the ODE. More critically, the classification 'nonlinear' is correct, but the reasoning 'y appears squared or multiplied by its own derivative' is a bit loose. Let's look closer. Is it linear? A linear ODE has the form a_n(x)y^(n) + ... + a_1(x)y' + a_0(x)y = g(x). Here we have y^2, so it is nonlinear. The statement 'It is nonlinear: y appears squared...' is correct. The 'or multiplied by its own derivative' part is just an additional example of nonlinearity, not a claim that it happens here. So the classification is correct. The verification is shown via an equation marked 'proved'. The sentence says 'Substituting ... makes the left side 0'. This is a correct verification. Why would this fail? Let's re-read carefully. 'Classify ... its order, whether it is linear and, if so, whether it is homogeneous.' The answer is 'order 1, nonlinear'. This is correct. The verification is correct. Is there a subtle error? The prompt says 'if so, whether it is homogeneous'. Since it's nonlinear, the homogeneous part doesn't apply. The solution doesn't mention homogeneity, which is fine. Wait, look at the verification equation: `Derivative(1/(4 - x), x) - 1/(4 - x)**2 = 0`. The ODE is `-y^2 + y' = 0`, which is `y' - y^2 = 0`. The substitution yields `y' = 1/(4-x)^2` and `y^2 = 1/(4-x)^2`. So `y' - y^2 = 0`. The equation in the solution is `Derivative(...) - ...**2 = 0`. This matches. It seems correct. Why 'fail'? Maybe the reasoning in step 2 is considered 'misleading' because it lists two conditions with 'or', implying either makes it nonlinear, which is true, but it's slightly imprecise phrasing? Or is there an error in the classification? No. Let's look at the 'homogeneous' part. Sometimes 'homogeneous' for nonlinear ODEs has a different meaning (homogeneous function), but standard calculus context usually reserves 'homogeneous' for linear ODEs or the specific form dy/dx = F(y/x). The problem asks 'if so [linear], whether it is homogeneous'. Since it's not linear, we don't need to check. The solution is sound. Is there a trick? 'Classify ... its order, whether it is linear ...'. The stated answer is 'order 1, nonlinear'. This is correct. The verification is correct. I will mark it pass. Wait, let me double check the 'nonlinear' reasoning. 'y appears squared or multiplied by its own derivative'. This is a sufficient condition for nonlinearity. It is not a necessary one. But it is a valid reason. I'll mark it pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims the ODE is nonlinear, but the equation y' - y^2 = 0 is actually nonlinear due to the y^2 term, so the classification is correct. However, the reasoning in step 2 is flawed: it states 'y appears squared OR multiplied by its own derivative'. While y^2 is present, the phrase 'or multiplied by its own derivative' suggests a condition for nonlinearity that is not met here (y is not multiplied by y'), and more importantly, the verification step is incomplete. The problem asks to verify the solution, but the solution only shows the algebraic check without explicitly stating that the substitution satisfies the ODE. More critically, the classification 'nonlinear' is correct, but the reasoning 'y appears squared or multiplied by its own derivative' is a bit loose. Let's look closer. Is it linear? A linear ODE has the form a_n(x)y^(n) + ... + a_1(x)y' + a_0(x)y = g(x). Here we have y^2, so it is nonlinear. The statement 'It is nonlinear: y appears squared...' is correct. The 'or multiplied by its own derivative' part is just an additional example of nonlinearity, not a claim that it happens here. So the classification is correct. The verification is shown via an equation marked 'proved'. The sentence says 'Substituting ... makes the left side 0'. This is a correct verification. Why would this fail? Let's re-read carefully. 'Classify ... its order, whether it is linear and, if so, whether it is homogeneous.' The answer is 'order 1, nonlinear'. This is correct. The verification is correct. Is there a subtle error? The prompt says 'if so, whether it is homogeneous'. Since it's nonlinear, the homogeneous part doesn't apply. The solution doesn't mention homogeneity, which is fine. Wait, look at the verification equation: `Derivative(1/(4 - x), x) - 1/(4 - x)**2 = 0`. The ODE is `-y^2 + y' = 0`, which is `y' - y^2 = 0`. The substitution yields `y' = 1/(4-x)^2` and `y^2 = 1/(4-x)^2`. So `y' - y^2 = 0`. The equation in the solution is `Derivative(...) - ...**2 = 0`. This matches. It seems correct. Why 'fail'? Maybe the reasoning in step 2 is considered 'misleading' because it lists two conditions with 'or', implying either makes it nonlinear, which is true, but it's slightly imprecise phrasing? Or is there an error in the classification? No. Let's look at the 'homogeneous' part. Sometimes 'homogeneous' for nonlinear ODEs has a different meaning (homogeneous function), but standard calculus context usually reserves 'homogeneous' for linear ODEs or the specific form dy/dx = F(y/x). The problem asks 'if so [linear], whether it is homogeneous'. Since it's not linear, we don't need to check. The solution is sound. Is there a trick? 'Classify ... its order, whether it is linear ...'. The stated answer is 'order 1, nonlinear'. This is correct. The verification is correct. I will mark it pass. Wait, let me double check the 'nonlinear' reasoning. 'y appears squared or multiplied by its own derivative'. This is a sufficient condition for nonlinearity. It is not a necessary one. But it is a valid reason. I'll mark it pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution claims the ODE is nonlinear, but the equation y' - y^2 = 0 is indeed nonlinear. However, the verification step is incomplete: it substitutes y into the equation but does not explicitly show that the result is identically zero for all x in the domain, nor does it address the domain of validity. More critically, the classification is correct (order 1, nonlinear), but the reasoning in step 2 is slightly imprecise ('y appears squared' is sufficient, 'or multiplied by its own derivative' is not present in this specific equation). The main issue is that the verification is presented as a single algebraic check without confirming it holds generally, which is a minor pedagogical gap, but the classification is correct. Wait, looking closer: the problem asks to classify AND verify. The solution classifies it as nonlinear (correct) and verifies the solution (correct). Is there an error? The equation is y' - y^2 = 0. This is nonlinear. The solution y = 1/(4-x) gives y' = 1/(4-x)^2 and y^2 = 1/(4-x)^2, so y' - y^2 = 0. The verification is correct. The classification is correct. Why would it fail? Ah, the prompt says 'if so, whether it is homogeneous'. Since it is nonlinear, the 'homogeneous' part of the linear classification doesn't apply in the standard linear sense, but nonlinear ODEs can be homogeneous in a different sense (degree of terms). Usually, 'homogeneous' in this context refers to linear homogeneous. The solution correctly identifies it as nonlinear. The verification is correct. Is there a subtle error? The solution says 'y appears squared OR multiplied by its own derivative'. In this equation, y is squared. It is NOT multiplied by its derivative. The 'or' makes the statement technically true (A or B is true if A is true), but it might be misleading if a student thinks the 'or' implies both conditions are possible reasons for nonlinearity in this specific case. However, the bigger issue is likely the verification. The equation provided in the solution is `Derivative(1/(4 - x), x) - 1/(4 - x)**2 = 0`. This is correct. Let's look at the classification again. Order 1: Correct. Nonlinear: Correct. Verification: Correct. Is it possible the question implies a linear homogeneous check? No, 'if so' covers that. I will mark it as pass unless I find a strict error. Wait, the solution text says 'y appears squared or multiplied by its own derivative'. This is a general statement about nonlinearity. It is not wrong. The verification is correct. I will pass it.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution claims the ODE is nonlinear, but the equation y' - y^2 = 0 is actually nonlinear because of the y^2 term, not because y is multiplied by its derivative. More importantly, the verification step is incomplete: it substitutes y into the equation but does not explicitly show the derivative calculation or confirm the result is identically zero for all x in the domain. However, the primary error is in the reasoning for nonlinearity: 'y appears squared' is correct, but 'or multiplied by its own derivative' is false for this specific equation (there is no y*y' term). The classification 'nonlinear' is correct, but the justification contains a factual error about the structure of the equation. Additionally, the problem asks to verify the solution, and step 3 is marked as an unchecked equation rather than a derived result, making the verification claim unsupported by the provided steps.
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.