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Mathematics Question on Differential equations

For the differential equations, find the general solution: x5dydx=−y5x^5\frac{dy}{dx}=-y^5

Answer

"The given differential equation is:
x5dydx=−y5x^5\frac{dy}{dx}=-y^5
⇒dyy5=−dxx5⇒\frac{dy}{y^5}=\frac{-dx}{x^5}
⇒dxx5+dyy5=0⇒\frac{dx}{x^5}+\frac{dy}{y^5}=0
Integrating both sides,we get:
∫dxx5+∫dyy5=k\int\frac{dx}{x^5}+∫\frac{dy}{y^5}=k(where k is any constant)
⇒∫x−5dx+∫y−5dy=k⇒∫x^{-5}dx+∫y^{-5}dy=k
⇒x−4−4+y−4−4=k⇒\frac{x^{-4}}{-4}+\frac{y^{-4}}{-4}=k
⇒x−4+y−4=−4k⇒x^{-4}+y^{-4}=-4k
⇒x−4+y−4=C    (C=−4k)⇒x^{-4}+y^{-4}=C\,\,\,\,(C=-4k)
This is the required general solution of the given differential equation.