Euler Equations having double roots solutions

Question

If the Euler equation have the double roots as a solution, second solution will be $y_2(x)=x^r\ln{x}$. What is its proof? or how it can be derived? To find a second solution,we will use the fact that constant times the solution is also a solution to linear homogeneous differential equations. Now why do we choose $\ln{x}$ as constant?why not any other constant?

Answer 1

One can find answer to this question here$\rightarrow$[Euler equation having double roots or repeated roots as a solution1DE]