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

Explanation

The equation of the family of ellipses having a center at the origin and foci on the x-axis is


`frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1` ......(i)


It is a two-parameter family of curves.


Differentiate equation (i) twice with respect to {x},


`frac{2 x}{a^{2}}+frac{2 y}{b^{2}} frac{d y}{d x}=0`


`frac{2}{a^{2}}+frac{2}{b^{2}}(frac{d y}{d x})^{2}+frac{2 y}{b^{2}} frac{d^{2} y}{d x^{2}}=0 . . .using frac{d(u . v)}{d x}=v frac{d u}{d x}+u frac{d v}{d x}`


Simplify the above equation,


`Rightarrow frac{x}{a^{2}}+frac{y}{h^{2}} frac{d y}{d v}=0 `……..(ii)


...