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Separable differential equations

Problem 6.98 · medium Video

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

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

Answer \( y = \sqrt{x^{2} + 4} \)

The slope field of y' = x/y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 2) is highlighted, y = sqrt(x² + 4).

Lines: 1 proved, 3 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 (style) — [domain objection, downgraded to style] The solution fails to address the sign ambiguity of the square root. While y(0)=2 selects the positive branch, the derivation steps are omitted, and the verification only checks the positive root, ignoring that y = -sqrt(x^2 + C) is also a solution to the ODE. A complete solution must justify why the negative root is discarded.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The solution fails to address the sign ambiguity of the square root. While y(0)=2 selects the positive branch, the derivation steps are omitted, and the verification only checks the positive root, ignoring that y = -sqrt(x^2 + C) is also a solution to the ODE. A complete solution must justify why the negative root is discarded.
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 final result against both the differential equation and the initial condition.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The solution fails to address the sign ambiguity when solving for y. Since y(0)=2 is positive, the correct branch is y = +sqrt(x^2 + C), but the text does not explicitly justify discarding the negative root, which is a critical step in initial-value problems involving square roots.
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.