The circles lie along?

The differential equation !!(dy)/(dx) = sqrt(1-y^2)/y!! determines a family of circles with:

  • Variable radii and a fixed centre at !!(0,1)!!
  • Variable radii and a fixed centre at !!(0,-1)!!
  • Fixed radius 1 and a variable centres along !!y-axis!!
  • Fixed radius 1 and a variable centres along !!x-axis!!

Solution:

We need to integrate the given differential equation and find the equation of the graph which will satisfy the given equation.

So we have !!ydy/(sqrt(1-y^2)) = dx!!

Now you could see that the differential of !!y^2!! is !!2y!!.

So we get !!-sqrt(1-y^2)=x+c!!

!!1-y^2=(x+c)^2!!

Hence the center of the circle lies on the x axis and the radius of the circle is 1.

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