Question

Form the differential equation of family of lines situated at a constant distance $p$ from the origin.

Answer 1

Equation of lines situated at a constant distance p from the origin, is given by,

$x\cos \theta +y\sin \theta =p$........(1)

differentiating on both sides w.r.t x, we get,

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

$\Rightarrow \cos \theta =-\frac{dy}{dx}\sin \theta$

substitute the above value in equation (1)

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

differentiating both sides of eqn (2) w.r.t x, we get,

$x\frac{d^{2}y}{d{x}^2}\sin \theta -\frac{dy}{dx}\sin \theta +\frac{dy}{dx}\sin \theta =0$

$x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+\frac{dy}{dx}=0$

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

Is the required differential equation.