Second Solution to Bessel's Function of order zero

[cybla]

Frobenius Method Exceptional case r1=r2

For the Frobenius Method for the exceptional case r1=r2... is the equation for the second solution


y$_{2}$= y$_{1}$ ln (x) + x$^{r_{1}+1}$$\sum_{n=0}^{\infty}b_{n}x^{n}$

or

y$_{2}$= y$_{1}$ ln (x) + x$^{r_{1}}$$\sum_{n=1}^{\infty}b_{n}x^{n}$

In a way both of them give the same exact answer however one begins with $b_{0}$x (the first one that begins at n=0) ...and the other begins with $b_{1}$x (the second one that begins at n=1)

Does it matter which one i use? Is one simpler than the other?

 

[cybla]

Disregard this post. I figured it out