Behaviour of Solutions to $x^2y'' + \alpha xy'+ \beta y = 0$ as $x \to 0$ and $x \to \infty$ 
Consider the Euler equation $x^2y'' + \alpha xy' + \beta y = 0$.
  Find conditions on $\alpha$ and $\beta$ so that:
  
  
*
  
*All solutions approach zero as $x \rightarrow 0$ 
  
*All solutions are bounded as $x \rightarrow 0$ 
  
*All solutions approach zero as $x \rightarrow \infty$ 
  
*All solutions are bounded as $x \rightarrow \infty$ 
  
*All solutions are bounded both as $x \rightarrow 0$ and $x \rightarrow \infty$
  

I am having trouble solving the problem. It must obviously have to do with the solutions to the indicial equation $ r^2 + (\alpha - 1)r + \beta= 0 \implies r = \dfrac{1 - \alpha
\pm \sqrt {(\alpha - 1)^2 - 4\beta^2}}{2} $. When these roots are identical or complex the solutions involve an $\ln x$ term the limit of which when $x \to 0$ is undefined I think? So my feeble argument is that the solutions for $r$ should be distinct to begin with? Please help. 

A logarithmic term, if present, only changes the behaviour as $x \to 0$ or $x \to \infty$ when $\text{Re}(r) = 0$.  That is, if $\text{Re}(r) \ne 0$ then both 
$x^r$ and $x^r \log(x)$ have the same limit or lack of limit as $x \to \infty$ and as $x \to 0$.  I don't know what that "or complex" is doing there: in the Euler equation a logarithmic term only appears when the roots are equal. 
