Question

Find the differential equation of the family of straight line at a fixed distance $P$ from the origin?

Answer 1

Let $x\cos \theta +y\sin \theta =p$ …………$\left(1\right)$ be the straight line

Differentiating wrt x, we get

$\cos \theta +\sin \theta \frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}=-\cot \theta$

(or) $\cos \theta =\frac{-dy}{dx}\left(\sin \theta \right)$

Substitute for $\cos \theta$ in $\left(1\right)$ we get

$x\left(\frac{-dy}{dx}\sin \theta \right)+y\sin \theta =p$ ……….$\left(2\right)$

Differentiate equation $\left(2\right)$ wrt x we get

$x\frac{d^{2}y}{d{x}^2}\left(\sin \theta \right)+(-1)\frac{dy}{dx}\left(\sin \theta \right)+\frac{dy}{dx}\left(\sin \theta \right)=0$

$\Rightarrow x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+\frac{dy}{dx}=0$ $(÷s\in \theta )$

$\Rightarrow x{d}^{2}y/d{x}^2=0$

$\Rightarrow x{y}^{\prime \prime }=0$ is the required differential equation.