If $\frac{dy}{dx}=\frac{2x-y+1}{x+2y+1},y\left(1\right)=1$,then $y\left(0\right)=$

$\frac{dy}{dx}=\frac{2x-y+1}{x+2y+1}$

$\Rightarrow$$(x+2y+1) d y+(y-2x-1) d x=0$

$\Rightarrow$$2 y d y-2 x d x+(x d y+y d x)+d y-d x=0$

$\Rightarrow$$y^2-x^2+xy+y-x=c$

$x=1,y=1\Rightarrow c=1$

$x=0\Rightarrow y^2+y-1=0\Rightarrow y=\frac{-1\pm \sqrt{5}}{2}$