Regarding the radial motion of photons

Photons move on null geodesics and the equation of motion on equatorial plane after some algebra can be written as

$$e^{nu}dot{t}^2-e^{-nu}dot{r}^2-r^2dot{phi}^2 = 0$$

$phi =0$ for the radial motion thus above equation becomes

$$e^{nu}dot{t}^2-e^{-nu}dot{r}^2 = 0$$

from where one can write down

$$frac{d r}{d t} = pm Big(1-frac{2M}{r}Big)$$

and integration will lead to

$$t = r +2MlnBig|1-frac{r}{2M}Big|+C hspace{12.5mm} And hspace{12.5mm} t = -r -2MlnBig|1-frac{r}{2M}Big|+C$$

Here I’m not sure what kind of physical meaning I should attribute to the final equations. I see at $r=2M$, $tto mpinfty$ but this also confuses my interpretation.