The equation of a circle, in general, unless anything is specified is,
x2 + y2 + 2gx + 2fy + c = 0 ----------eqn (1)
The above equation contains three constants which cannot be further reduced.
So, the order of the differential equation should be 3.
Differentiating both sides of (1) w.r.t x, we have,
2x + 2y + 2g + 2f y1 = 0 i.e x + y + g + f y1 = 0
Again diff,
1 + y1 + f y2 = 0 i.e f = - ( 1 + y1 ) / y2
Diff, once more,
y2 + f y3 = 0 i.e y2 - ( 1 + y1 ) y3 / y2 = 0
i.e (y2)2 = ( 1 + y1 ) y3
The above is the differential equation of a circle x2 + y2 + 2gx + 2fy + c = 0
Cheers !!
Moderation Team
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