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Separable differential equations

Problem 6.93 · medium

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 4 \).

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.
\[ \frac{d}{d x} \sqrt{2 x^{2} + 16} = \frac{2 x}{\sqrt{2 x^{2} + 16}} \]
The solution satisfies the equation.✓ Proved
\[ 4 \]
And the initial condition.✓ Proved

Answer \( y = \sqrt{2 x^{2} + 16} \)

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 solution skips the actual derivation steps (separation of variables, integration, and solving for the constant) and instead only verifies the final answer. This fails to demonstrate the method required to solve the initial-value problem.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution skips the actual derivation steps (separation of variables, integration, and solving for the constant) and instead only verifies the final answer. This fails to demonstrate the method required to solve the initial-value problem.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to address the sign ambiguity inherent in solving for y (y = ±sqrt(...)). While the positive root is correct for this specific initial condition, the method described ('solve for y') is incomplete without explicitly justifying the choice of sign based on y(0)=4 > 0.
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.