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Separable differential equations

Problem 6.48 · medium Video

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

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

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

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, 1) is highlighted, y = sqrt(x² + 1).

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 fails to address the sign ambiguity inherent in solving for y (y = ±sqrt(x^2 + C)). While the positive root is correct for this specific initial condition, the derivation steps described do not explicitly justify discarding the negative root, which is a critical logical step in solving separable differential equations involving square roots.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to address the sign ambiguity inherent in solving for y (y = ±sqrt(x^2 + C)). While the positive root is correct for this specific initial condition, the derivation steps described do not explicitly justify discarding the negative root, which is a critical logical step in solving separable differential equations involving square roots.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the method of separation of variables and verifies the result against the differential equation and initial condition. Although the explicit integration steps are omitted, the logical flow and final verification are sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The solution fails to address the sign ambiguity inherent in solving y^2 = x^2 + C. While y(0)=1 selects the positive branch, the steps do not explicitly justify discarding the negative root, which is a critical part of solving separable equations involving squares.
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.