The differential equation of all circles which passes through the origin and whose centre lies on y axis is

Dear pallavi

let the center is (o,a)  and  radius is r

so equation of circle is

x² + (y-a)² =r²

 differentiate

2x +2(y-a) dy/dx =0

x+ (y-a)dy/dx =0  ..............1

again differentiate

1+ (y-a)d²y/dx² + dy/dx =0

put value of (y-a) from equation 1

1 -x(d²y/dx²)/(dy/dx) + dy/dx =0

(dy/dx )² + dy/dx  - xd²y/dx²=0

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