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Separable differential equations

Problem 6.96 · medium Video

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

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

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

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

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 claims to solve for y but fails to determine the sign of the square root. Since y(0) = 5 > 0, the solution must be y = +sqrt(x^2 + 25); the negative root is extraneous. The provided steps do not explicitly justify discarding the negative branch, which is a critical part of solving for y explicitly.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The solution claims to solve for y but fails to determine the sign of the square root. Since y(0) = 5 > 0, the solution must be y = +sqrt(x^2 + 25); the negative root is extraneous. The provided steps do not explicitly justify discarding the negative branch, which is a critical part of solving for y explicitly.
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 result against both the differential equation and the initial condition. The steps are logically sound and complete.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the method of separation of variables and applies the initial condition to determine the constant. The final answer is correct and verified.
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.