Question

Form the differential equation of the family of hyperbolas having foci on x -axis and centre at origin.

Answer 1

It is given that the hyperbolas have center at origin and foci on x-axis.

So, the equation of hyperbolas is,

x 2 a 2 − y 2 b 2 =1

Here a and b are arbitrary constant.

Differentiate the above equation with respect to x,

d dx ( x 2 a 2 − y 2 b 2 )= d dx ( 1 ) 2x a 2 − 2y y ′ b 2 =0 x a 2 − y y ′ b 2 =0 (1)

Again differentiate above equation with respect to x,

d dx ( x a 2 − y y ′ b 2 )=0 1 a 2 − 1 b 2 [ y ′ y ′ +y y ″ ]=0 1 a 2 = 1 b 2 [ y ′ y ′ +y y ″ ]

Substitute 1 a 2 = 1 b 2 [ y ′ y ′ +y y ″ ] to equation (1),

x b 2 [ ( y ′ ) 2 +y y ″ ]− y y ′ b 2 =0 x ( y ′ ) 2 +xy y ″ −y y ′ =0 xy y ″ +x ( y ′ ) 2 −y y ′ =0

Therefore, the differential equation of family of hyperbolas is xy y ″ +x ( y ′ ) 2 −y y ′ =0.