Question

Form the differential equation of the family of circles touches the X-axis at the origin.

Answer 1

Since the circle touches the $x-axis$ at origin.

The center will be on the $y-axis$

So,

$x$ - coordinate of the center

i.e.,

$a=0$

since,center=$(0,b)$

So,

equation of circle

$=>(x-0{)}^2+(y-b{)}^2=b^2$

$=>x^2+(y-b{)}^2=b^2$

$=>x^2+y^2+b^2-2yb=b^2$

$=>x^2+y^2=2yb--------------\left(i\right)$

Since,there is variable we differentiate once

Differentiate w.r.t $x$ we get

$=>2x+2y\frac{dy}{dx}=2b\times \frac{dy}{dx}$

$=>x+y{y}^{\prime }=b{y}^{\prime }$

$=>b=\left[\frac{x+y{y}^{\prime }}{y^{\prime }}\right]$

Putting the value of b in $\left(i\right)$ we get

$=>x^2+y^2=2y\left[\frac{x+y{y}^{\prime }}{y^{\prime }}\right]$

$=>(x^2+y^2)y^{\prime }=2y(x+y{y}^{\prime })$

$=>x^2{y}^{\prime }+y^2{y}^{\prime }=2yx+2y^2{y}^{\prime }$

$=>y^{\prime }(x^2-y^2)=2xy$

$=>y^{\prime }=\frac{2xy}{x^2-y^2}$