How to say $ a_n(r) $ is arbitrary in Frobenius method?

Question

Usually $ a_0(r) $ in Frobenius method is arbitrary. However when the instance is the difference of two roots is positive integer, there will be two cases:

1. When trying out the smaller root, two arbitrary constant are found. The solution is non-logarithmic.
2. When trying out the smaller root, only one arbitrary constant is found. The solution is logarithmic.

How can you say that $ a_n(r_1) $ is arbitrary before getting recurrence relations?

Let $ r_1 $ the smaller root.