find the differential equation of the family of curves (x-h)^2+(y-k)^2=r^2, where h and k are arbitrary constants - Maths - Differential Equations

Question

find the differential equation of the family of curves
(x-h)^2+(y-k)^2=r^2, where h and k are arbitrary constants

Lovina Kansal · Accepted Answer

Dear student
The equation of the family of circles of radius r isx-h2+y-k2=r2    
(1)where h and h are parameters
Since equation(1) contains two arbitrary constants, we differentiate it two times w
r
t xand the didderential equation will be of second order
Differentiating (1) wrt x, we get2(x-h)+2(y-k)dydx=0x-h+(y-k)dydx=0      
(2)Differentiating (2) w r t
x, we get1+y-kd2ydx2+dydx2=0    
(3)From (3), we havey-k=-1+dy/dx2d2y/dx2      
(4)Putting the value of y-k in (2), we obtainx-h=1+dydx2dydxd2y/dx2Substituting the value of x-h and y-k in (1), we get1+dydx22dydx2d2y/dx22+1+dy/dx22d2y/dx22=r21+dy/dx23=r2d2y/dx22This is the required differential equation
Regards