<s>[INST] How do I solve this equation for R/r? How do I solve this expression for $\frac{R}{r}$?
$\frac{1}{(r-R)^3}-\frac{y}{R^2(r-R)}=\frac{1}{r^3}$
 [/INST] Rearrange the equation to get
$$1 - y \left ( 1 - \frac{r}{R} \right )^2 = \left ( 1 - \frac{r}{R} \right )^3 $$
Set $z = 1-r/R$ and get the equation:
$z^3+y z^2-1=0$
Get a root at $z_0(y)$ numerically or otherwise.  Then $R/r$ is then
$$\frac{R}{r} = \frac{1}{1-z_0(y)} $$
NB There is an analytical expression for $z_0$:
$$ z_0(y) = \frac{1}{3} \left(\frac{\sqrt[3]{-2 y^3+3 \sqrt{3} \sqrt{27-4
   y^3}+27}}{\sqrt[3]{2}}+\frac{\sqrt[3]{2} y^2}{\sqrt[3]{-2 y^3+3 \sqrt{3}
   \sqrt{27-4 y^3}+27}}-y\right) $$
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