Show that the given differential equation is homogeneous and solve each of them.`x^2(dy)/(dx)=x^2-2y^2+x y`

Explanation

`x^(2)dy-(x^(2)+xy-2y^(2))dx=0`
`implies (dy)/(dx)=(x^(2)+xy-2y^(2))/(x^(2))`
`implies v+x(dv)/(dx)=(x^(2)+x^(2)v-2x^(2)v^(2))/(x^(2))`
`=1+v-2v^(2)`
`implies x(dv)/(dx)=-1-2v^(2)`
`implies (dv)/(1-2v^(2))=(dx)/(x)`
Let `y=vx`
`implies (dy)/(dx)=v+x(dv)/(dx)`
`implies (1)/(2)int(1)/(((1)/(sqrt(2)))^(2)-v^(2))dv=int(dx)/(x)+c`
`implies (1)/(2)*(1)/(2*(1)/(sqrt(2)))log"((1)/(sqrt(2))+v)/((1)/(sqrt(2))-v)=logx+c`
`implies (1)/(2sqrt(2))log"(1+vsqrt(2))/(1-vsqrt(2))=logx+c`
`implies(1)/(2sqrt(2))log"(1+(y)/(x)sqrt(2))/(1-(y)/(x)sqrt(2))=logx+c`
`implies (1)/(2sqrt(2))log"(x+ysqrt(2))/(x-ysqrt(2))=logx+c`