Case $1$ : $\alpha=0$
Then $\ddot{r}r=0$
$r=0$ or $\ddot{r}=0$
$r=0$ or $r=C_1t+C_2$
$\therefore r=C_1t+C_2$
Case $2$ : $\alpha\neq0$
Then $\ddot{r}r=\alpha(\dot{r}^2-1)$
$r\dfrac{d^2r}{dt^2}=\alpha\left(\left(\dfrac{dr}{dt}\right)^2-1\right)$
Let $u=\dfrac{dr}{dt}$ ,
Then $\dfrac{d^2r}{dt^2}=\dfrac{du}{dt}=\dfrac{du}{dr}\dfrac{dr}{dt}=u\dfrac{du}{dr}$
$\therefore ru\dfrac{du}{dr}=\alpha(u^2-1)$
$\dfrac{u}{u^2-1}du=\dfrac{\alpha}{r}dr$
$\int\dfrac{u}{u^2-1}du=\int\dfrac{\alpha}{r}dr$
$\dfrac{1}{2}\ln(u^2-1)=\alpha\ln r+c_1$
$\ln\left(\left(\dfrac{dr}{dt}\right)^2-1\right)=2\alpha\ln r+c_2$
$\left(\dfrac{dr}{dt}\right)^2-1=C_1r^{2\alpha}$
$\dfrac{dr}{dt}=\pm\sqrt{C_1r^{2\alpha}+1}$
$dt=\pm\dfrac{dr}{\sqrt{C_1r^{2\alpha}+1}}$
$\int dt=\pm\int\dfrac{dr}{\sqrt{C_1r^{2\alpha}+1}}$
$t=\pm\int_k^r\dfrac{dr}{\sqrt{C_1r^{2\alpha}+1}}+C_2$