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

Explanation

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.