Solutions of a partial differential equations

I'm looking for solutions of a PDE of the form : $$P(t,x):[0,\infty]*[0,b]\rightarrow[0,1]$$ $$\partial_t P(t,x)= \partial_x [(1+2x) P(t,x)]$$ $$P(0,x)=\delta_x (0)$$ $$P(t,b)=g(t)$$

Where $$b$$ is a boundary for $$x$$ and $$g(t)$$ is a known function. Is there anyone who knows how to solve this?

• Change of variable $y=\log (2x+1)$? – Nemo Feb 17 at 6:32