Two differential equations representing the family of circles touching \(y-\) axis at the origin is:

Linear and of the first order

circle touching y axis at origin

$(x-a)^2+y^2=a^2$

$x^2-2ax+y^2=0$  ...(1)

differentiating wrt x

$2x-2a+2yy'=0$

$x-a+yy'=0$

$x+yy'=a$  ...(2)

putting 'a' from (2) in (1)

$x^2-2(x+yy')x+y^2=0$

$x^2-2x^2-2xyy'+y^2=0$

$y^2-x^2-2xyy'=0$ linear first order