How to solve first order non-linear ODE $y'=\frac{2x+y+4}{4x-2y}$

Question

I can solve first order non-linear ODE like below: $$y'=\frac{2x+y+4}{4x+2y}$$ This is quite easy by substituting $2x+y$ with $u$. However, I have tried to solve the differential equation as title: $$y'=\frac{2x+y+4}{4x-2y}$$ Note that the denominator has the negative sign $4x-2y%$. It seems that neither substituting nor setting $v=\frac{y}{x}$ is working. Please help me with the detailed answer. Thanks a lot.

Answer 1

Hint:

In this case let $x=u+\alpha$ and $y=v+\beta$ with substitution obtain suit $\alpha$ and $\beta$ which equation to be homogeneous. $$y'=\frac{2x+y+4}{4x-2y}$$ $$v'=\frac{2u+2\alpha+v+\beta+4}{4u+4\alpha-2v-2\beta}$$ \begin{cases} 2\alpha+\beta+4=0,\\ 4\alpha-2\beta=0. \end{cases} $\alpha=-1$ and $\beta=-2$. so solve $$v'=\frac{2u+v}{4u-2v}$$