Solution of the differential equation xdx+ydyxdx−ydy=y3x3 is given by (Base of all log=e )

We have xdx+ydyxdx−ydy=y3x3⇒dx32+dy32dx32−dy32=y32x32⇒du+dvdu−dv=vu Where u=x32 and v=y32⇒udu+udv=vdu−vdv⇒udu+vdv=vdu−udv⇒udu+vdvu2+v2=vdu−udvu2+v2⇒du2+v2u2+v2=−2dtan−1⁡vu On integrating we get log⁡u2+v2=−2tan−1⁡vu−2c⇒12log⁡x3+y3+tan−1⁡yx32+c=0