Question

Form the differential equation of the family of circles having centre on y -axis and radius 3 units.

Answer 1

It is given that the circle have the centre on y-axis and radius 3 units.

So, the equation of circles is,

x 2 + ( y−b ) 2 = 3 2 x 2 + ( y−b ) 2 =9 (1)

Differentiate above equation with respect to x,

d dx [ x 2 + ( y−b ) 2 ]= d dx ( 9 ) 2x+2( y−b ) dy dx =0 x+( y−b ) y ′ =0 y−b= −x y ′

Substitute y−b= −x y ′ in equation (1),

x 2 + ( −x y ′ ) 2 =9 x 2 + x 2 ( y ′ ) 2 =9 x 2 ( 1+ 1 ( y ′ ) 2 )=9 x 2 [ ( y ′ ) 2 +1 ]=9 ( y ′ ) 2

Simplify further,

x 2 ( y ′ ) 2 −9 ( y ′ ) 2 + x 2 =0 ( y ′ ) 2 ( x 2 −9 )+ x 2 =0

Therefore, the differential equation of family of circles is ( y ′ ) 2 ( x 2 −9 )+ x 2 =0.