For the differential equation in given question find the general solution.

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

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Solution

Given, $x^{5} \frac{d y}{d x} = - y^{5}$
On separating the variables, we get $\frac{d y}{y^{5}} = - \frac{d x}{x^{5}} O n i n t e g r a t i n g, w e g e t \int \frac{d y}{y^{5}} = - \int \frac{d x}{x^{5}} \Rightarrow y^{- 5} d y = - \int x^{- 5} d x$
\Rightarrow $\frac{y^{- 5 + 1}}{(- 5 + 1)} = - \frac{x^{- 5 + 1}}{(- 5 + 1)} + C$ $(∵ \int x^{n} d x = \frac{x^{n + 1}}{n + 1})$
(\Rightarrow $\frac{y^{- 4}}{(- 4)} = - \frac{x^{- 4}}{(- 4)} + C$ \Rightarrow $x^{- 4} + y^{- 4} = - 4 C$
\Rightarrow $x^{- 4} + y^{- 4} = A [w h e r e, A = - 4 C]$
which is the required general solution.

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