Frobenius Method - Roots differ by integer

[asras]

I'm reading up on some methods to solve differential equations. My textbook states the following:

"$y_{1}$ and $y_{2}$ are linearly independent ... since $\sigma_{1}-\sigma_2$ is not an integer."

Where $y_{1}$ and $y_{2}$ are the standard Frobenius series and $\sigma_1$ and $\sigma_2$ are the roots of the indicial equation.

I'm having trouble seeing how the above follows and would appreciate some input. I'm using "Essential Mathematical Methods for the Physical Sciences" and the quote is (albeit slightly paraphrased) from page 282, for reference.

Incidentally this is my first post. Looking forward to participating in this forum.

 

[Somenath]

y₁ and y₂ are linearly dependent if ay₁ + by₂ = c
a,b,c= const.
let, y₁=$\sum$_nx^n+σ₁
y₁=$\sum$_nx^n+σ₁
then ay₁$\sum$_nx^n+σ₁ + by₁$\sum$_nx^n+σ₁=c

Since right hand side is const. , all the coefficients of the powers of x are zero. This is possible if one term arising in the first summation cancels the other. This is possible only when σ1 and σ2 differ by integer. then n can assume different values and cancel the coefficients.
So, for independentness, σ1-σ2=fraction