When solving second-order differential equations by the Frobenius method at a regular singular point, you are supposed to use the two roots of the indicial equation to give you two independent solutions.
If there is only one root, it makes sense that you would need another method to get the second independent solution. However, many texts say that you also need to do this when the roots differ by an integer.
Why?
Is it that when the roots differ by an integer, the two matching solutions are not independent? If so, why must they be independent?
Is it that sometimes they will be independent and sometimes they won't? If so, when will they be independent and when won't they?
Is it that there is something that prevents calculating one of the solutions? If so, why?
Best Answer
If you look at http://math.creighton.edu/nielsen/DE_Fall_2010/Series%20Solutions/Series_Solutions_Beamer.pdf, they write the resulting recurrence for one of the solutions as $a_n F(n+r) = E$, where $F(r)$ is the indicial equation, and I'm writing $E$ to abbreviate a complicated expression which depends on a variety of things, including $n$. If $F(n+r)=0$ for some $n$ and $E \neq 0$, then this isn't solvable.