simplify your first derivative before going further
notice you can divide each term by 1/3 to get
x^(-2/3) + y^(-2/3) dy/dx = 0
dy/dx = -x^(-2/3) / y^(-2/3)
= - y^(2/3)/x^(2/3)
= - (y/x)^(2/3)
d^2y/dx^2 = (-2/3) (y/x)^(-1/3) [( xdy/x - y)/x^2)
replace the dy/dx in the square bracket by (y/x)^2/3) and see what you get.
Still messy but a small improvement.