I am called to solve the following differential equation
$$F'''(r)+\frac{\partial\log R(r)}{\partial r}F''(r)+ \frac{\partial^2\log R(r)}{\partial r^2} F'(r)=0$$
with $r\in(-\infty,0]$. The solution, which I found with the help of Wolfram Alpha, is given below
$$F(r)=c_0+\int_0^r dr'\frac{1}{R(r')} \Big[ c_1+c_2\int_0^{r'} dr'' R(r'')\Big]$$
where $c_i,\ i=0,1,2$ are constants and the primes do not indicate differentiation. The function $R(r)$ is related to the integration variable with the following way $$dr=-\frac{dR}{1-(\frac{R}{R_+})^4}$$ with $R\in[0,R_+]$. So, changing variables from $r$ to $R$ yields
$$F(R)=c_0-\int_0^R dR'\frac{1}{R'[1-(\frac{R'}{R_+})^4]}\bigg( c_1-c_2\int_0^{R'} dR'' \frac{R''}{[1-(\frac{R''}{R_+})^4]}\bigg)$$
The last integral can be evaluated, yielding
$$F(R)=c_0-\int_0^R dR'\frac{1}{R'[1-(\frac{R'}{R_+})^4]}\bigg( c_1-c_2\frac{R_+^2}{4}\ln\Big[\frac{R_+^2+R'^2}{R_+^2-R'^2}\Big]\bigg)$$
Now, I wish to impose B.C.s on the solution $F(R)$. I wish 1) my solution to be constant at $R=R_+$ and 2) my solution to be finite at $R=0$. Those are two B.C.s only, so I realize that there will be left one out of the three constants in the final solution after imposing the B.C.s.
The obvious choice is to set $c_1=c_2=0$, but I would like first to exclude any other possibilities before arriving to this conclusion. So, my question is whether or not there is any other possibility that meets my two requirements. For instance, demanding the solution to be finite at $R=0$ may fix $c_0$ as $$c_0=\int_0^{R_0} dR'\frac{1}{R'[1-(\frac{R'}{R_+})^4]}\bigg( c_1-c_2\frac{R_+^2}{4}\ln\Big[\frac{R_+^2+R'^2}{R_+^2-R'^2}\Big]\bigg)$$ and therefore the remaining solution to be $$F(R)=c_0-\int_{R_0}^R dR'\frac{1}{R'[1-(\frac{R'}{R_+})^4]}\bigg( c_1-c_2\frac{R_+^2}{4}\ln\Big[\frac{R_+^2+R'^2}{R_+^2-R'^2}\Big]\bigg)$$ and last, but not least, to determine $c_2$ as a function of $c_1$ by requiring $F(R)$ to be constant at $R_+$ (which I do not see how that happens)... Can this line of reasoning be performed completely or a similar line of reasoning to be adopted, or am I forced to conclude that $c_1=c_2=0$?