Frobenius power series repeated roots

[John777]

Could someone please explain the y₂ solution for repeated roots in Frobenius method where y₂=y₁lnx+x^s $$\Sigma$$ C_nxⁿI am struggling to figure out how to solve this

 

[matematikawan]

For a solution about a regular singular point x=0, look at simplest case first in the form of Euler-Cauchy equation
$$x^2y''+\alpha xy'+\beta y=0$$

when the indicial root is a double root.