The solution of the differential equation x−y  arc   tanyxdx+x  arcTanyxdy=0 is of the form logx4x2+y2+k tan−1yx   = c then the value of k is

The given equation can rewritten as dydx=ytan−1⁡yx−xxtan−1⁡yx=yx−1tan−1⁡y/xv+xdvdx=v−1tan−1⁡v ( on substituting y=vx)⇒∫tan−1⁡vdv=−∫dxxvtan−1⁡v−∫v11+v2dv+ln⁡x=c   by parts⇒ln⁡x−12∫2v1+v2dv+yxtan−1⁡yx=c⇒2ln⁡x−ln⁡1+v2+2yxtan−1⁡yx=c⇒2ln⁡x−ln⁡x2+y2x2+2yxtan−1⁡yx=c⇒ln⁡x4x2+y2+2yxtan−1⁡yx=c