Show that the differential equation `y^(2)dx+(x^(2)-xy+y^(2))dy=0` is homogenous and solve it.

Show that the differential equation `y^(2)dx+(x^(2)-xy+y^(2))dy=0` is

1.	Show that the differential equation `y^(2)dx+(x^(2)-xy+y^(2))dy=0` is homogenous and solve it.
Answer» The given differential equation may be written as `(dy)/(dx)=-y^(2)/(x^(2)-xy+y^(2))`……………..(i) On dividing the Nr and Dr of RHS of (i) by `x^(2)`, we get `(dy)/(dx)=-(y/x)^(2)/(1-y/x+(y/x)^(2))=f(y/x)`. Thus, the given differential equation is homogeneous. Putting, `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `v+x(dv)/(dx) =(-v^(2))(1-v+v^(2))` `rArr x(dv)/(dx)-v^(2)/(1-v+v^(2)+v]` `rArr x(dv)/(dx)=-(v+v^(3))/(1-v+v^(2))` `rArr (1-v+v^(2))/(v(1+v^(2)))dv=-1/xdx` `rArr int((1+v^(2))-v)/(v(1+v^(2)))dv=-int(dx)/x`. `rArr int(dv)/v-int(dv)/(1+v^(2))+int(dx)/x=logC` `rArr log\|v\|-tan^(-1)v+log\|x\|=logC` `rArr tan^(-1)v=log(\|vx\|)/C` `rArr tan^(-1)(y/x)=log(\|y\|/C) [therefore v=y/x]` `rArr \|y\|/c=e^(tan^(-1)(y//x))` `rArr \|y\|=Ce^(tan^(-1)(y//x))`, which is the required solutions.