What would be the logical mistake if indicial equation roots are actually different by an integer, but still mathematically one solves the equation following the route of non-integer difference, still evaluated at ordinary singular point? Suppose,the ODE is of second order. Would doing so might result in two linearly independent solutions whose one particular linear combination in a terminating series? I am following textbook on Ordinary and Partial differential equations by Dr. M.D. Raisinghania, but logic behind the method is not mentioned in the book.
I don't have access to the book you mentioned, I could have been more specific with my response. I will implore you to work it out yourself to see why it is that way; this will guide you to fully understand the art behind the statement not just because said / write so. If you need help with it, let me know.
If the roots are of the indicial polynomial differer by an integer (including zero) the Frobenius method only gives one root. See
https://en.wikipedia.org/wiki/Frobenius_method
for details of both cases - roots don't differer by integer and they do differer by an integer.
The logic is that we need to find two independent solutions. If they differ by an integer, one can be obtained from others by setting few zero coefficients on series. You can find the proof of it from Kreyszig Advanced Engineering Mathematics book (Appendix 4)
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