Question

From the differential equation of the family of circles touching the y-axis at the origin.

Answer 1

The Centre of the circle touching the y-axis at origin lies on the x-axis.
Let (a,0) be the Centre of the circle.
Since, it touches the y-axis at origin, its radius is a.
Now, the equation of the circle with Centre (a,0)and radius a is
$(x-a)^2+y^2=a^2$.
$\Rightarrow x^2+y^2=2ax$...(1)

Differentiating equation (1)with respect to x, we get:
2x+2yy'=2$a$
$\Rightarrow$ x+yy'= $a$
Now, on substituting the value of a in equation(1),we get:
x2+y2=2(x+yy')x
$\Rightarrow$ x2+y2=2x2+2xyy'
$\Rightarrow$ 2xyy'+x2=y2

This is the required differential equation.