The solution of frac{dy}{dx} = left( frac{y}{ | vedclass

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Question

(C) Given the differential equation: $\frac{dy}{dx} = \left( \frac{y}{x} \right)^{1/3}$.Separating the variables,we get: $\frac{dy}{y^{1/3}} = \frac{d

Vedclass · Accepted Answer

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

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(C) Given the differential equation: $\frac{dy}{dx} = \left( \frac{y}{x} \right)^{1/3}$.Separating the variables,we get: $\frac{dy}{y^{1/3}} = \frac{dx}{x^{1/3}}$.Integrating both sides: $\int y^{-1/3} dy = \int x^{-1/3} dx$.Using the power rule for integration $\int x^n dx = \frac{x^{n+1}}{n+1} + c$,we get:$\frac{y^{2/3}}{2/3} = \frac{x^{2/3}}{2/3} + c$.Multiplying by $\frac{2}{3}$,we obtain: $y^{2/3} = x^{2/3} + c'$,where $c'$ is a constant.Rearranging the terms,we get: $y^{2/3} - x^{2/3} = c$.

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

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