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Q.
The differential equation of the family of ellipse having foci on $Y$-axis and centre at origin is
Differential Equations
Solution:
The equation of family of ellipse is of the form
$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1 ....$(i)
(since, foci is on $Y$-axis, so we draw a vertical ellipse)
On differentiating Eq. (i) w.r.t. x, we get
$ \frac{d}{d x}\left(\frac{x^2}{b^2}\right)+\frac{d}{d x}\left(\frac{y^2}{a^2}\right) =\frac{d}{d x}(1) .....$(ii)
$\Rightarrow \frac{1}{b^2} 2 x+\frac{1}{a^2} 2 y y^{\prime} =0$
$\frac{y y^{\prime}}{x} =-\frac{a^2}{b^2}$
Again, differentiating w.r.t. $x$, we get
$\Rightarrow \frac{x \frac{d}{d x}\left(y y^{\prime}\right)-y y^{\prime} \frac{d}{d x}(x)}{x^2}=0$
$\left[\right.$ using uqotient rule $\left.\frac{d}{d x}\left(\frac{u}{v}\right)=\frac{v \frac{d}{d x}(u)-u \frac{d}{d x}(v)}{v^2}\right]$
$\Rightarrow \frac{x\left[y y^{\prime \prime}+\left(y^{\prime}\right)^2\right]-y y^{\prime} \cdot 1}{x^2}=0$
$\left[\right.$ using product rule $\left.\frac{d}{d x}(u \cdot v)=\left(u \frac{d}{d x} v+v \frac{d}{d x} u\right)\right]$
$\Rightarrow x\left(y^{\prime}\right)^2+x y y^{\prime \prime}-y y^{\prime}=0 $
$\Rightarrow x y y^{\prime \prime}+x\left(y^{\prime}\right)^2-y y^{\prime}=0$
which is the required differential equation.
