Question

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

Answer 1

Equation of family of ellipse is of the form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ ...(i)
[Since, foci on Y-axis, so we draw a vertical sllipse]
On differentiating Eq. (i) w.r.t. x, we get
$\frac{d}{dx}\left(\frac{x^2}{b^2}\right)+\frac{d}{dx}\left(\frac{y^2}{a^2}\right)=\frac{d}{dx}\left(1\right)...\left(ii\right)$
$\Rightarrow \frac{1}{b^2}2x+\frac{1}{a^2}2y{y}^{\prime }=0\Rightarrow \frac{y{y}^{\prime }}{x}=-\frac{a^2}{b^2}$
Again differentiating w.r.t. x, we get
$\Rightarrow \frac{x\frac{d}{dx}\left(y{y}^{\prime }\right)-y{y}^{\prime }\frac{d}{dx}\left(x\right)}{x^2}=0$
(using quotient rule $\frac{d}{dx}\left(\frac{u}{v}\right))=\frac{v\frac{d}{dx}\left(u\right)-u\frac{d}{dx}\left(v\right)}{v^2}\Rightarrow \frac{x[y{y}^{\prime \prime }+(y^{\prime }{)}^2]-y{y}^{\prime }.1}{x^2}=0$
Using product rule $\frac{d}{dx}(u.v)=(u\frac{d}{dx}v+v\frac{d}{dx}u)\Rightarrow x(y^{\prime }{)}^2+xy{y}^{\prime \prime }-y{y}^{\prime }=0\Rightarrow xy{y}^{\prime \prime }+x(y^{\prime }{)}^2-y{y}^{\prime }=0$
which is the required diffrential equation.