Question

Find the differential equation of all the ellipse whose center at origin and axis are along the coordinate axis.

Answer 1

An ellipse whose center is at origin and the axes are the coordinate axis is represented by the equation.

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a>b$ -(i)

Differentiating (i) with respect to $x$ once we obtain

$\frac{2x}{a^2}+\frac{2y{y}^{{}^{\prime }}}{b^2}=0$ where $y^{{}^{\prime }}=\frac{dy}{dx}$ -(ii)

Differentiating (ii) with respect to $x$ we obtain

$\frac{2}{a^2}+2\left[\frac{{(y^{{}^{\prime }})}^2+y^{{}^{\prime \prime }}}{b^2}\right]=0$ where $y^{{}^{\prime \prime }}=\frac{d^{2}y}{{dx}^2}$

$\Rightarrow \frac{b^2}{a^2}=\frac{-[{(y^{{}^{\prime }})}^2+y^{{}^{\prime \prime }}]}{1}$ -(iv)

Consider equation (ii)

$\frac{2x}{a^2}+\frac{2y{y}^{{}^{\prime }}}{b^2}=0\Rightarrow \frac{x}{a^2}+\frac{y{y}^{{}^{\prime }}}{b^2}=0\Rightarrow \frac{b^2}{a^2}=-\frac{y{y}^{{}^{\prime }}}{x}$ -(v)

Equating (iv) & (v) we have

$\frac{y{y}^{{}^{\prime }}}{x}={y^{{}^{\prime }}}^2+y^{{}^{\prime \prime }}\Rightarrow y{y}^{{}^{\prime }}=x[{\left(y^{{}^{\prime }}\right)}^2+y^{{}^{\prime \prime }}]\Rightarrow y\frac{dy}{dx}=x[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{{dx}^2}]$