There are a few special cases- in particular, "Cauchy-Euler type equations" have coefficient of each a constant times : .
(These are also called "equi-potential equations" because the power of x is the same as the order of the derivative.)
The substitution t= ln(x) changes that type of equation to a "constant coefficients" equation.
But the most general method is to look for a power series solution. That is, write . Find the derivatives of that, put them into the equation (If the coefficients are not polynomials, write Taylor series for them), and you typically can get a "recursion relation" for the numbers .
If the coefficient of the highest order derivative is 0 at the point at which you have your initial values, then you may need to use "Frobenius' method": Frobenius Method -- from Wolfram MathWorld
with something of the form
where c is a constant, not necessarily positive or an integer.