The solution of frac{dy}{dx} = (frac{x}{y})^{ | vedclass

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Question

(B) Given the differential equation: $\frac{dy}{dx} = (\frac{x}{y})^{-1/3}$.By simplifying the right side,we get: $\frac{dy}{dx} = (\frac{y}{x})^{1/3}

Vedclass · Accepted Answer

$y^{2/3} - x^{2/3} = c$

Vedclass · Answer

(B) Given the differential equation: $\frac{dy}{dx} = (\frac{x}{y})^{-1/3}$.By simplifying the right side,we get: $\frac{dy}{dx} = (\frac{y}{x})^{1/3} = \frac{y^{1/3}}{x^{1/3}}$.Separating the variables,we have: $y^{-1/3} dy = x^{-1/3} dx$.Integrating both sides:$\int y^{-1/3} dy = \int x^{-1/3} dx$.Using the power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$:$\frac{y^{2/3}}{2/3} = \frac{x^{2/3}}{2/3} + C_1$.Multiplying by $\frac{2}{3}$:$y^{2/3} = x^{2/3} + \frac{2}{3}C_1$.Let $c = \frac{2}{3}C_1$,then $y^{2/3} - x^{2/3} = c$.

The solution of $\frac{dy}{dx} = (\frac{x}{y})^{-1/3}$ is

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