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

Explanation

The given differential equation may be written as
`(dy)/(dx)=(x^(2)-2y^(2)+xy)/(x^(2))`.
On dividing the Nr and Dr of RHS of (i)by `x^(2)`, we get
`(dy)/(dx)={1-2(y/x)^(2)+(y/x)}=f(y/x)`.
So, the given differential equation is homogenous.
Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get
`v+x(dv)/(dx)=(x^(2)-2v^(2)x^(2)+vx^(2))/(x^(2))`
`rArr v+x(dv)/(dx)=1-2v^(2)+v`
`rArr x(dv)/(dx)=1-2v^(2)`
`rArr int (dv)/(1-2v^(2))=int(dx)/x`.
`rArr 1/2int(dv)/(1-2v^(2))=int(dx)/x`.
`rArr 1/2int(dv)/({(1/sqrt(2))^(2)-v^(2)})=int(dx)/x`.
`rArr 1/2.1/(2 xx 1/sqrt(2))log|(1/sqrt(2)+v)/(1/sqrt(2)-v)|=log|x|+C`
`rArr 1/(2sqrt(2))log|(x+sqrt(2)y)/(x-sqrt(2)y)|-log|x|=C [therefore v=y/x]`.
This is the required solution.