Question

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

Answer 1

Finding General solution
Given differential equation is

$x^5\frac{dy}{dx}=-y^5$

$\frac{dy}{y^5}=\frac{-dx}{x^5}$

Integrating both sides
We get,

$\int \frac{dy}{y^5}=\int \frac{-dx}{x^5}$

$\frac{y^{-5+1}}{-5+1}=\frac{-x^{-5+1}}{-5+1}+c$

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

$x^{-4}+y^{-4}+4c=0$

taking $4c=k$

$x^{-4}+y^{-4}+K=0$

Final Answer:
Hence, the required general solution is

$x^{-4}+y^{-4}+K=0$