Question

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

Answer 1

`y'=(2y-x)/(2x-y)`

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


`xv'+v=(2xv-x)/(2x-xv)`

`xv'=(2x(v-1))/(x(2-v))-v`

`xv'=(2(v-1))/(2-v)-(v(2-v))/(2-v)`

`xv'=(v^2-2)/(2-v)`

This is separable, so you have


`(2-v)/(v^2-2)\,\mathrm dv=\frac{\mathrm dx}x`

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