Let \(y\left(x\right)\) be the general solution of \(x^{5}\frac{dy}{dx}=-y^5\). Let \(y\left(1\right)=1\) then

\(\frac{1}{x^4}-\frac{1}{y^4}=1\) \(\frac{1}{x^4}+\frac{1}{y^4}=2\) \(\frac{1}{x^4}=1+\frac{1}{y^3}\) \(\frac{1}{x^4}-\frac{1}{y^4}=2\)

\(\frac{1}{x^4}+\frac{1}{y^4}=2\)

Given differential equation is separable