Discussion Forum

The differential equation representing the family of circles of constant radius r is 

Answer:

Explanation:

Family of circle of constant radius r is 

 $(x-a)^{2}+(y-b)^{2}=r^{2}$

 Let $x=a + r \cos \theta$,y=$b+r \sin \theta$

 $\frac{dx}{d\theta}=-r \sin \theta, \frac{dy}{d \theta}= r \cos \theta \Rightarrow \frac{dy}{dx}=-\cot \theta$

 $\frac{dy ^{2}}{dx^{2}}=cosec ^{2} \theta, \frac{d\theta}{d x}= \frac{- cosec ^{2} \theta}{r}$

$\frac{dy ^{2}}{dx^{2}}= \frac{-(1+ \cot ^{2} \theta)^{3/2}}{r}$

 $ry"= -(1+(y')^{2})^{3/2}$

 squaring on both sides , we get

 $r^{2} (y")^{2}=[1+(y')^{2}]^{3}$