Question

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

Answer 1

It is given that ellipses have center at origin and foci on y-axis.

So, the equation of ellipse is,

x 2 b 2 + y 2 a 2 =1

Differentiate the above equation with respect to x,

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

Again differentiate above equation with respect to x,

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

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

x[ − 1 a 2 ( y ′ ) 2 +y y ″ ]+ y y ′ a 2 =0 −x ( y ′ ) 2 −xy y ″ +y y ′ =0 xy y ″ +x ( y ′ ) 2 −y y ′ =0

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