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

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

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