Question

Show that the differential equation of the system of parabola $y^2=4a(x-b)$ is given by

Answer 1

The numbers of constants, is equal to the number of time we differentiate.

Here, there are two constants, so we differentiate twice.

$y^2=4a(x-b)$

$\Rightarrow 2y\frac{dy}{dx}=4a$

$\Rightarrow y\frac{dy}{dx}=\frac{2a}{y}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=2a\frac{d}{dx}\left(\frac{1}{y}\right)$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-2a}{y^2}$

$\Rightarrow y^2\frac{d^{2}y}{d{x}^2}+2a=0$

Hence $y^2\frac{d^{2}y}{d{x}^2}+2a=0$ is the required differential equation.