Question

For the differential equations, find the general solution: \(x^5\frac{dy}{dx}=-y^5\)

Answer 1

"The given differential equation is:
\(x^5\frac{dy}{dx}=-y^5\)
\(⇒\frac{dy}{y^5}=\frac{-dx}{x^5}\)
\(⇒\frac{dx}{x^5}+\frac{dy}{y^5}=0\)
Integrating both sides,we get:
\(\int\frac{dx}{x^5}+∫\frac{dy}{y^5}=k\)(where k is any constant)
\(⇒∫x^{-5}dx+∫y^{-5}dy=k\)
\(⇒\frac{x^{-4}}{-4}+\frac{y^{-4}}{-4}=k\)
\(⇒x^{-4}+y^{-4}=-4k\)
\(⇒x^{-4}+y^{-4}=C\,\,\,\,(C=-4k)\)
This is the required general solution of the given differential equation.