A.Math

(b^2)(x^2)-(a^2)(y^2)=(a^2)(b^2)

Differentiate both sides with respect to x

2(b^2)x-2(y^2)y(dy/dx)=0

dy/dx=(b^2x/a^2y)

Differentiate both sides with respect to x

d^2y/dx^2=[a^2yb^2-b^2xa^2(dy/dx)] / (a^4y^2)

d^2y/dx^2=[a^2b^2y-a^2b^2(dy/dx) / (a^4y^2)

d^2y/dx^2=[a^2b^2y-a^2b^2(b^2x/a^2y)] / (a^4y^2)

d^2y/dx^2=[a^2b^2y-b^2(b^2x/y)] / (a^4y^2)

d^2y/dx^2=(a^2b^2y^2-b^4x^2) / (a^4*y^3)

2x b^2 - 2y a^2 dy/dx = 0 ---------(1)
dy/dx = (x b^2) / (y a^2) ------------(2)

from (1)
2 b^2 - 2 a^2 [ dy/dx dy/dx + y ( d^2y/dx^2 ) ] = 0
2 b^2 - 2 a^2 ( dy/dx )^2 - 2 a^2 y ( d^2y/dx^2 ) = 0

from (2)
2 b^2 - 2 a^2 [ (x b^2) / (y a^2) ]^2 - 2 a^2 y ( d^2y/dx^2 ) = 0

simply, the answer can be found