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

Explanation

The general equation of an ellipse having foci on the x-axis and centre at the origin, is given by
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1" "…(i),` where a and b are the parameters.
Since this equation contains two parameters, so we shall differentiate it twice to get the required differential equation.

Differentiating (i) w.r.t. x, we get:
`(2x)/(a^(2))+(2y)/(b^(2)).(dy)/(dx)=0implies(y)/(b^(2)).(dy)/(dx)+(x)/(a^(2))=0`
`implies(yy_(1))/(b^(2))=(-x)/(a^(2)), where (dy)/(dx)=y_(1)`
`implies(yy_(1))/(x)=(-b^(2))/(a^(2))." "...(ii)`
Differentiating (ii) w.r.t.x, we get
`(x.(d)/(dx)(yy_(1))-yy_(1).(d)/(dx)(x))/(x^(2))=0`
`impliesx[yy_(2)+(y_(1))^(2)]-yy_(1)=0`
`implies(xy)y_(2)+x(y_(1))^(2)-yy_(1)=0.`
Hence, `(xy)(d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)-y((dy)/(dx))=0` is the required differential equation.