Find the general solution of the differential equation x^ 5 d y d… - Vidyadip

Question

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

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Answer

Given differential equation is: $x^{5} \frac{d y}{d x}=-y^{5}$
Separating the variables we get
$ \frac{d y}{y^{5}}=\frac{-d x}{x^{5}}$
Integrating both sides,
$\Rightarrow \int \frac{d y}{y^{5}}= -\int \frac{d x}{x^{5}}$,
$\Rightarrow \int \mathrm{y}^{-5} \mathrm{dy}= -\int \mathrm{x}^{-5} \mathrm{dx}$
$\Rightarrow$ $\frac{y^{-4}}{-4}=-\frac{x^{-4}}{-4}+C_{1}$
$\Rightarrow$ $-y^{-4}=x^{-4}+4 c_{1}$
$\Rightarrow x^{-4}+y^{-4}=c$

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