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Question:

$x^{5} \frac{d y}{d x}=-y^{5}$

Solution:

The given differential equation is:

$x^{5} \frac{d y}{d x}=-y^{5}$

$\Rightarrow \frac{d y}{y^{5}}=-\frac{d x}{x^{5}}$

 

$\Rightarrow \frac{d x}{x^{5}}+\frac{d y}{y^{5}}=0$

Integrating both sides, we get:

$\int \frac{d x}{x^{5}}+\int \frac{d y}{y^{5}}=k \quad$ (where $k$ is any constant)

$\Rightarrow \int x^{-5} d x+\int y^{-5} d y=k$

$\Rightarrow \frac{x^{-4}}{-4}+\frac{y^{-4}}{-4}=k$

$\Rightarrow x^{-4}+y^{-4}=-4 k$

$\Rightarrow x^{-4}+y^{-4}=\mathrm{C}$                   $(\mathrm{C}=-4 k)$

This is the required general solution of the given differential equation.

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