Repeated complex conjugate roots for Cauchy-Euler

[N@te]

Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation.

This is incorrect, but I think it is close:
X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2]

I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?

 

[BvU]

Repeated roots doesn't mean you only have a c1 and a c2. See Maslanka top of p 3.

 

[N@te]

That is the Rosetta stone I needed. Thanks. My auxiliary equation is wrong which is throwing everything else off.

 

[BvU]

You're most welcome. And I learned too from digging in this (after all, not every physicist can offhandedly cough up what a 4th order CE eqn is, so I had to google too )