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

Explanation

The equation of the family of ellipse whose foci are on `Y`-axis and the centre is at origin, is
`(x^(2))/(b^(2))+(y^(2))/(a^(2))=1`………..`(1)`

differentiate w.r.t.x,
`(d)/(dx)((x^(2))/(b^(2)))+(d)/(dx)((y^(2))/(a^(2)))=(d)/(dx)(1)`..................`(2)`
`implies (1)/(b^(2))2x+(1)/(a^(2))2yy'=0implies(yy')/(x)=-(a^(2))/(b^(2))`
Again differentiate w.r.t.x
`implies`(x(d)/(dx)(yy')-yy'(d)/(dx)(x))/(x^(2))=0`
`implies(x[yy''+(y')^(2)]-yy'*1)/(x^(2))=0`
`implies xyy''+x(y')^(2)-yy'=0`
which is the required differential equation.