Is x 0 an ordinary or a singular point of the differential

Solution

0 is ordinary point of a(x)y\'\'+c(x)y=0 if c(x)/a(x) is analytic at 0
c(x)/a(x)=sin(x)/x
sin(x)=_k=0.. (-1)^k/(2k+1)!*x^2k+1

so sin(x)/x=_k=0.. (-1)^k/(2k+1)!*x^2k

sin(x) is convergent for any x, hence the above series for sin(x)/x converges for any x

So sin(x)/x has an interval of convergence around 0 ,(-,) so sin(x)/x is analytic