Question

How do i solve this differential equation y'=2y-x over 2x-y

Answer 1

$y'=\dfrac{2y-x}{2x-y}$

Let $y=xv$, where $v=v(x)$, so that $y'=xv'+v$. Then the ODE is

$xv'+v=\dfrac{2xv-x}{2x-xv}$
$xv'=\dfrac{2x(v-1)}{x(2-v)}-v$
$xv'=\dfrac{2(v-1)}{2-v}-\dfrac{v(2-v)}{2-v}$
$xv'=\dfrac{v^2-2}{2-v}$

This is separable, so you have

$\dfrac{2-v}{v^2-2}\,\mathrm dv=\dfrac{\mathrm dx}x$

Integrate both sides, solving for $v$ if possible, then replace using $v=\dfrac yx$ and solve for $y$ if possible.