McCoy13
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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.
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.