Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation? I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its physical application. Can anybody please give an example (relevant in physics) of constructing a series solution in such a case about infinity, in particular if infinity is a regular singular point?
Second, what does it mean to develop a series solution about "infinity"? I guess infinity is not very well-defined and unique like other points on finite axis, like x=0. So how to make sense of such a series?
 A: I will try to provide examples where a solutions of physical problems are expansions about $\infty$. They are not necessarily solutions to a second-order differential equation. 

Mechanics: 
Consider a cylinder of mass M, area A and length $L_0$. Situate it vertically on the ground,  (that is, one end touching the ground, the other end in the air). Suppose that the cylinder (like every solid object) has elasticity, that is, spring constant, equal to K. Now, if the cylinder is infinitely rigid, that is, if K is equal to infinity, the length of the cylinder will not change due to the gravitation, and will be equal to $L_0$. That is, gravity will not make it shrink, since K is infinity. 
But what if the cylinder is rigid, but some elasticity (like every real solid)? Then we expect the length to become less than L, we expect it to shrink a bit. So, we seek a solution in the form: $L=L_0 - \frac{1}{k} f_1(M,g,A,L_0)$, where $f_1$ is the first-order correction in terms of $\frac{1}{K}$. You can continue to find the next corrections in powers of $\frac{1}{K}$. 

Electrostatics: This time in electrostatics. You can certainly find the electric field above the center of a plane with uniform surface density $\sigma$. It is $\frac{\sigma}{  2 \epsilon}$. Now, suppose that instead of the infinite plane, which physically does not exist, we have a circular disk of radius $R$, ans we try to find the electric field on its axis, at distance $h$ from the center. You can easily calculate the result, which is $\frac{\sigma}{2\epsilon} \bigg[ 1-\frac{h}{h^2+R^2}\bigg]$. Since $R$ is large, you can Taylor-expand this in terms of $\frac{h}{R}$. The result is $\frac{\sigma}{  2 \epsilon} -\frac{\sigma}{  2 \epsilon} \frac{h}{R}+ \frac{\sigma}{  4  \epsilon} \frac{h^3}{R^3} + \ldots $. This is an expansion about $R=\infty$. 

thermodynamics:  almost every occurrence of the term 'finite size effects' in thermodynamics, in phase transitions and in the field of complex networks would give you corrections (which are basically truncated expansions) in powers of the inverse of system size. 

Finally, I think I can contrive an illustrative example of second order ordinary differential equations where there is a singularity at $\infty$. Suppose there is a block  long rectangular block  of mass M, on the ground, with no friction between the block and the ground. Suppose we place a mass $m$ on the block, with friction $\mu$ between the blocks. Suppose that $M$ is large, that is, $m \ll M$. Now we pull the upper mass with constant force $F$. Find the velocity of the block (not the upper mass) as a function of time. If you write down equations of motion, you will see that for $M=\infty$, there is a singularity, easily removable, and the speed (or position) of the block is zero for $M=\infty$, and the corrections are of order $\frac{1}{M}$. 
