The solution of the differential equation $\left(1-x y-x^5 y^5\right) d x-x^2\left(x^4 y^4+1\right) d y=0$ is given by

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

The given equation is $d x-x(y d x+x d y)=x^5 y^4(y d x+x d y)$

$\Rightarrow \frac{d x}{x}=\left(1+x^4 y^4\right) d(x y) \Rightarrow \ln x=x y+\frac{1}{5} x^5 y^5+\ln c$

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

Hence (1) is the correct answer.