Linear System of ODEs: Solving for n=1 or n=3

[McCoy13]

Homework Statement
I'm trying to solve the following system of ODEs.

$$\alpha = \alpha (r)$$

$$\alpha ' + \frac{n-1}{2r} \alpha =0$$

$$\alpha '' + \frac{n-1}{r} \alpha ' = 0$$


The attempt at a solution

The solution to the first one is

$$\alpha = r^{\frac{-(n-1)}{2}$$

The solution to the second one is

$$\alpha '= r^{-(n-1)}$$

Ultimately the goal is to show that n=1 or n=3 (it's a problem dealing with wave attenuation and distortion, but I'm just having problems with this step). I really can't reconcile these answers, even using arbitrary scalar factors against my solutions. When I tried substituting one equation into the other all that happened was I ended up with a factor of sqrt(2) that wasn't consistent with either equation individually.

 

[fzero]

The solutions to these equations do match when n=3, just be a bit more careful with the algebra. The equations themselves are not well-behaved when n=1, so you should be careful when trying to match the solutions for arbitrary n to the limit n->1. It happens to work for the first equation, but there is an additional solution of the second that is not a solution to the first.

 

[McCoy13]

Gah, I see. I think I was forgetting that r was to a negative power when I was thinking about it in the case n = 3 and in the case n = 1 I wasn't looking at the equations, just the solutions, so I missed the fact that they sort of become singular. Thanks for elucidating.