Separable differential equations
Problem 6.122 · medium Video
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.Reviewed
- Use y(0) to find the constant of integration, then solve for y.Reviewed
- \[ \frac{d}{d x} 2 e^{- 3 x} = - 6 e^{- 3 x} \]The solution satisfies the equation.✓ Proved
- \[ 2 \]And the initial condition.Reviewed
Answer \( y = 2 e^{- 3 x} \)
Lines: 1 proved, 3 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly outlines the method of separation of variables and verifies the final answer against the differential equation and initial condition. The steps are logically sound and mathematically accurate.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly outlines the method of separation of variables and verifies the final answer against the differential equation and initial condition. The steps are logically sound and mathematically accurate.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-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 2026-09-27qwen3.6:27b-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 tgpt-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 2026-10-13 with SymPy 1.14.0.