# Solve this following

Question:

The differential equation satisfied by the system of parabolas $\mathrm{y}^{2}=4 \mathrm{a}(\mathrm{x}+\mathrm{a})$ is :

1. $y\left(\frac{d y}{d x}\right)^{2}-2 x\left(\frac{d y}{d x}\right)-y=0$

2. $y\left(\frac{d y}{d x}\right)^{2}-2 x\left(\frac{d y}{d x}\right)+y=0$

3. $y\left(\frac{d y}{d x}\right)^{2}-2 x\left(\frac{d y}{d x}\right)+y=0$

4. $y\left(\frac{d y}{d x}\right)+2 x\left(\frac{d y}{d x}\right)-y=0$

Correct Option: , 3

Solution:

$y^{2}=4 a x+4 a^{2}$

differentiate with respect to $x$

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

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

so, required differential equation is

$\mathrm{y}^{2}=\left(4 \times \frac{\mathrm{y}}{2} \frac{\mathrm{dy}}{\mathrm{dx}}\right) \mathrm{x}+4\left(\frac{\mathrm{y}}{2} \frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}$

$\Rightarrow \mathrm{y}^{2}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}+2 \mathrm{xy}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}^{2}=0$

$\Rightarrow \mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}+2 \mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}=0$