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Separable differential equations

Problem 6.8 · medium Video

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 5 \).

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
\[ \frac{d}{d x} \sqrt{2 x^{2} + 25} = \frac{2 x}{\sqrt{2 x^{2} + 25}} \]
The solution satisfies the equation.✓ Proved
\[ 5 \]
And the initial condition.Reviewed

Answer \( y = \sqrt{2 x^{2} + 25} \)

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, 5) is highlighted, y = sqrt(2x² + 25).

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: pass
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — 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 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution skips the actual derivation steps (separation of variables, integration, and solving for the constant) and only provides a verification. A solution to an initial-value problem must show how the answer is obtained, not just that it is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (misleading) 2026-09-26 — The solution skips the actual derivation steps (separation of variables, integration, and solving for the constant), presenting only a verification. This fails to demonstrate the method required to solve the problem.
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.