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Answer: \(3xy^2\dfrac{dy}{dx} + 2x^3 - y^3 = 0\)

Given \(x^3 + y^3 = 35ax\) ... (1), with one arbitrary constant \(a\).

Differentiating w.r.t. \(x\): \(3x^2 + 3y^2\dfrac{dy}{dx} = 35a\) ... (2)

From (1), \(35a = \dfrac{x^3 + y^3}{x}\). Substituting in (2): \(x\left(3x^2 + 3y^2\dfrac{dy}{dx}\right) = x^3 + y^3\)

\(\therefore 3x^3 + 3xy^2\dfrac{dy}{dx} - x^3 - y^3 = 0\), i.e. \(3xy^2\dfrac{dy}{dx} + 2x^3 - y^3 = 0\), which is the required differential equation.

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report
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