Why do we use $x^{m_1}$ instead of $e^{m_1x}$ for general solutions to Cauchy-Euler equations? When I learned to calculate the general solution to homogeneous linear differential equations, I was told that for say,
$y''-y=0$, the auxiliary equation is $m^2-1=0\implies m^2=1\implies m=\pm 1$ and so the solution is $y=c_1e^{m_1x}+c_2e^{m_2 x}=c_1e^{x}+c_2e^{-x}$.
However, for Cauchy-Euler differential equations, which are linear and are homogeneous, the solutions take the form $y=c_1x^{m_1}+c_2x^{m_2}$.
This is the case for which the auxiliary equation provides two distinct real roots. 
My question is if both the Cauchy-Euler equations and the non-Cauchy-Euler equations are homogeneous and they are both linear, why do one of the solutions involve $e^{m_1x}$ and the other involves just $x^{m_1}$? Why can't I just use $e^{m_ix}$ to find the solution even for the Cauchy-Euler ones?
 A: Tiny answer: because it works (and other things don't). 
Short answer: they are actually the same solutions. It's just that $x$ that you see in the Cauchy-Euler equation corresponds to $e^x$ in the constant-coefficient equation. The Cauchy-Euler equation is the constant-coefficient equation in disguise. (The disguise being a change of independent variable.)
Long answer: your perception has been shaped by the unbalanced diet of textbook examples, which consists of equations for which certain methods work.  For any randomly chosen linear homogeneous equation of second order, such as
$$ y'' + e^x y' + x y=0 \tag{1}$$
(which I just made up), those methods do not work, even if you try different substitutions for $y$ all day long. We simply cannot describe these solutions (other than $0$)  in terms of functions for which we already have a name.  Unless, of course, you create a new name for the solution - which is what people do (Bessel functions, Airy functions, etc). It seems nobody gave a name to the solutions of (1); you can be first. 
The general situation with differential equations is that no clever substitutions can produce a formula for the solution. There are rare cases when a miracle happens (an explicit solution is obtained); you should appreciate those unique miracles, not expect such occasions to be frequent and uniform.   
