Question

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

Answer 1

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