I am looking for a radial solution on $B_1(0) \subset \mathbb{R}^2$. $$\Delta v=v''(r)+ {1 \over r}v'(r)=r-1$$
I already know that $v(r)=c_1 \ln(r)+c_2$ is a solution for $\Delta v=0$ and $v(1)=0$.
Setting $y=v'$ and $y'=v''$ we reach a linear equation of first degree: $$y'= r-1-{y \over r}$$ How can this be continued?