# The differential equation satisfied by the system of parabolas

Question:

The differential equation satisfied by the system of parabolas

$y^{2}=4 a(x+a)$ is:

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

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

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

4. (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 $x$

$\Rightarrow 2 y \frac{d y}{d x}=4 a$

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

so, required differential equation is

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

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

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

Leave a comment

Click here to get exam-ready with eSaral