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

.: ${x^5}\cfrac{{dy}}{{dx}} = - {y^5} \Rightarrow \cfrac{{dy}}{{{y^5}}} = - \cfrac{{dx}}{{{x^5}}}$

…(1)
Integrating (1) both sides,

we get
$\int {{y^{ - 5}}dy} = - \int {{x^{ - 5}}} dx\; \Rightarrow \cfrac{{{y^{ - 4}}}}{{ - 4}} = - \cfrac{{{x^{ - 4}}}}{{ - 4}} + {C_1}$

$\Rightarrow \cfrac{{{x^{ - 4}}}}{4} + {C_1} = \cfrac{{ - {y^{ - 4}}}}{4} \Rightarrow \cfrac{{{x^{ - 4}}}}{4} + \cfrac{{{y^{ - 4}}}}{4} = - {C_1} \Rightarrow {x^{ - 4}} + {y^{ - 4}} = C$

$[C = - 4{C_1}]$

which is required solution.