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Question

(A) The given differential equation is: $x^{5} \frac{dy}{dx} = -y^{5}$ Separating the variables,we get: $\frac{dy}{y^{5}} = -\frac{dx}{x^{5}}$ Rearran

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$x^{-4} + y^{-4} = C$

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(A) The given differential equation is: $x^{5} \frac{dy}{dx} = -y^{5}$ Separating the variables,we get: $\frac{dy}{y^{5}} = -\frac{dx}{x^{5}}$ Rearranging the terms: $\frac{dx}{x^{5}} + \frac{dy}{y^{5}} = 0$ Integrating both sides: $\int x^{-5} dx + \int y^{-5} dy = k$ (where $k$ is an arbitrary constant) Applying the power rule for integration $\int x^{n} dx = \frac{x^{n+1}}{n+1}$: $\frac{x^{-4}}{-4} + \frac{y^{-4}}{-4} = k$ Multiplying by $-4$: $x^{-4} + y^{-4} = -4k$ Let $C = -4k$,then the general solution is: $x^{-4} + y^{-4} = C$

Find the general solution of the differential equation: $x^{5} \frac{dy}{dx} = -y^{5}$

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