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Separable differential equations

Problem 6.47 · medium Video

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

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.
\[ \frac{d}{d x} 2 e^{x^{2}} = 4 x e^{x^{2}} \]
The solution satisfies the equation.✓ Proved
\[ 2 \]
And the initial condition.

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

The slope field of y' = 2x·y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 2) is highlighted, y = 2e^(x²).

Lines: 1 proved, 3 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
4	Not checked	—	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: pass
qwen3.6:27b-mlx: fail (error) — The solution is incomplete because it fails to show the actual derivation steps (separation of variables, integration, and solving for the constant). It only provides the final answer and verifies it, which does not demonstrate how to solve the problem.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution is incomplete because it fails to show the actual derivation steps (separation of variables, integration, and solving for the constant). It only provides the final answer and verifies it, which does not demonstrate how to solve the problem.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims to separate variables and integrate, but fails to show the integration step or the determination of the constant C. It jumps from the method description to the final answer without deriving it, making the 'solution' incomplete and logically disconnected.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the method of separation of variables and verifies the result against the differential equation and initial condition. The steps are logically sound and complete.
gpt-oss:20b: pass 2026-09-26

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.