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Separable differential equations

Problem 6.122 · medium

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 2 \).

Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
Use y(0) to find the constant of integration, then solve for y.
\[ \frac{d}{d x} 2 e^{- 3 x} = - 6 e^{- 3 x} \]
The solution satisfies the equation.✓ Proved
\[ 2 \]
And the initial condition.✓ Proved

Answer \( y = 2 e^{- 3 x} \)

Lines: 2 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.

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
4	✓ 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	sympy.checkodesol substitutes the solution into the equation; y(0) matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The provided equation check is incorrect: the derivative of 2*exp(-3*x) is -6*exp(-3*x), but the differential equation requires dy/dx = -3y = -6*exp(-3*x). Wait, -3 * (2*exp(-3*x)) = -6*exp(-3*x). So the equation actually holds. Let me re-read carefully. dy/dx = -3y. y = 2e^(-3x). dy/dx = -6e^(-3x). -3y = -3(2e^(-3x)) = -6e^(-3x). They match. The initial condition y(0)=2 holds. The steps described are correct. Why would it fail? Ah, looking at the JSON output format required. The prompt asks to find what is wrong. Is there anything wrong? The solution is correct. The steps are standard. The checks are correct. Wait, let me look closer at the 'proved' lines. Line 3 says 'Derivative(2*exp(-3*x), x), -6*exp(-3*x)'. This is just calculating the derivative. It doesn't explicitly state that this equals -3y. However, the comment says 'The solution satisfies the equation.' This is a valid check if one assumes the reader verifies -6exp(-3x) == -3(2exp(-3x)). Is this misleading? No. Is it an error? No. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. The solution has sentences and equations. The sentences describe the method. The equations verify the answer. The answer is correct. The method is correct. The verification is correct. I should pass it.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The provided equation check is incorrect: the derivative of 2*exp(-3*x) is -6*exp(-3*x), but the differential equation requires dy/dx = -3y = -6*exp(-3*x). Wait, -3 * (2*exp(-3*x)) = -6*exp(-3*x). So the equation actually holds. Let me re-read carefully. dy/dx = -3y. y = 2e^(-3x). dy/dx = -6e^(-3x). -3y = -3(2e^(-3x)) = -6e^(-3x). They match. The initial condition y(0)=2 holds. The steps described are correct. Why would it fail? Ah, looking at the JSON output format required. The prompt asks to find what is wrong. Is there anything wrong? The solution is correct. The steps are standard. The checks are correct. Wait, let me look closer at the 'proved' lines. Line 3 says 'Derivative(2*exp(-3*x), x), -6*exp(-3*x)'. This is just calculating the derivative. It doesn't explicitly state that this equals -3y. However, the comment says 'The solution satisfies the equation.' This is a valid check if one assumes the reader verifies -6exp(-3x) == -3(2exp(-3x)). Is this misleading? No. Is it an error? No. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. The solution has sentences and equations. The sentences describe the method. The equations verify the answer. The answer is correct. The method is correct. The verification is correct. I should pass it.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The verification step in line 3 is mathematically incorrect: the derivative of 2*exp(-3*x) is -6*exp(-3*x), but the differential equation requires t
gpt-oss:20b: pass 2026-09-27

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/separable_ode, checked 2026-09-27 with SymPy 1.14.0.