From the differential equation $$\frac{\partial P}{\partial r} = \left[1+\frac{r}{\ln(1+r)}\right]D$$
I get the second-order equation
$$\frac{1}{D}{P(r)}=\text{Ei}\left(2\ln(r+1)) - \text{Ei}(\ln(r+1)\right) + c_{2}r + c_{1}$$
and the following Lagrangian
$$L(P',P,r) = \frac{1}{2}\left(P'^2+2P\left(\frac{-r}{(r+1)\ln^2(r+1)}+\frac{r}{(r+1)\ln(r+1)}+\frac{1}{(r+1)\ln(r+1)}\right)\right).$$
I would like to see the stable (and unstable) points by plotting the Euler-Lagrange solutions with a contour or density plot. Can someone give me advice - preferably in Mathematica or R - on how to do this?
Thanks. SH