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# Find the differential equation that will represent the family of all circles having centres on the X-axis and the radius is unity.

Toolbox:
• If we have an equation $f(x,y,c_1,c_2,....c_n)=u$ Containing n arbitrary constant $c_1,c_2...c_n$, then by differentiating n times, we get $(n+1)$ equations in total. If we eliminate the arbitrary constants $c_1,c_2....c_n,$ we get a D.E of order n
$(x-a)^2+y^2=1$ -----(i)
It represents the family of circles with center $(a,0)$ on the x axis and radius =1
$2(x-a)+2y \large\frac{dy}{dx}=0$-----(ii)
Step 2:
from (ii) $(x-a)=-y \large\frac{dy}{dx}$
Substitute in (i)