Question

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

Answer 1

The equation of the family of ellipses having centre at the origin and foci on the x-axis is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ...(1)
where $a\mathrm{and}b$ are the parameters.
As this equation contains two parameters, we shall get a second-order differential equation.
Differentiating (1) with respect to $x$, we get
$\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0$ ...(2)
Differentiating (2) with respect to $x$, we get
$$\begin{gathered} \frac{2}{a^2}+\frac{2}{b^2}\left[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}\right]=0 \\ \Rightarrow \frac{2}{a^2}=-\frac{2}{b^2}\left[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}\right] \\ \Rightarrow \frac{b^2}{a^2}=-\left[{\left(\frac{dy}{dx}\right)}^2+y\left(\frac{d^{2}y}{d{x}^2}\right)\right]...\text{(3)} \end{gathered}$$
Now, from (2), we get
$$\begin{gathered} \frac{x}{a^2}=-\frac{y}{b^2}\frac{dy}{dx} \\ \Rightarrow \frac{b^2}{a^2}=-\frac{y}{x}\frac{dy}{dx}...\left(4\right) \end{gathered}$$
From (3) and (4), we get
$$\begin{gathered} -\frac{y}{x}\frac{dy}{dx}=-\left[{\left(\frac{dy}{dx}\right)}^2+y\left(\frac{d^{2}y}{d{x}^2}\right)\right] \\ \Rightarrow \frac{y}{x}\frac{dy}{dx}=\left[{\left(\frac{dy}{dx}\right)}^2+y\left(\frac{d^{2}y}{d{x}^2}\right)\right] \\ \Rightarrow y\frac{dy}{dx}=x{\left(\frac{dy}{dx}\right)}^2+xy\left(\frac{d^{2}y}{d{x}^2}\right) \\ \Rightarrow xy\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-y\frac{dy}{dx}=0 \\ \text{It is the required differential equation.} \end{gathered}$$