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Separable differential equations

Problem 6.94 · medium

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 1 \).

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
\[ \frac{d}{d x} \sqrt{2 x^{2} + 1} = \frac{2 x}{\sqrt{2 x^{2} + 1}} \]
The solution satisfies the equation.✓ Proved
\[ 1 \]
And the initial condition.✓ Proved

Answer \( y = \sqrt{2 x^{2} + 1} \)

Lines: 2 proved, 2 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.

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	✓ 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: pass — The solution correctly identifies the method of separation of variables and verifies the result against the differential equation and initial condition. The logic is sound and complete.

Every verdict on record (4)

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 logic is sound and complete.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims to separate variables and integrate, but provides no integration steps or derivation of the constant. It merely verifies the final answer, failing to demonstrate the solution process requested by 'Solve'.
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 2026-09-26 with SymPy 1.14.0.