this method works, modified, on any problem which can be factored into first order operators, and where one can solve first order problems. another example is the so called Eulers equation.
Similarly for Euler's equation, x^2y'' +(1-a-b)xy' + ab y = 0, with
(r-a)(r-b) = 0, just factor x^2y'' +(1-a-b)xy' + ab y = (xD-a)(xD-b)y = 0,
and solve (xD-a)z = 0, and then (xD-b)y = z.
As above, this proves existence and uniqueness simultaneously, and also
handles the equal roots cases at the same time, with no guessing.
Here you have to use, I guess, integrating factors to solve the first order acses, and be careful when "multiplying" the non constant coefficient operators (xD-a), since you must use the leibniz rule.
these are usually done by powers series methods, or just stating that the indicial equation should be used, again without proving there are no other solutions. of course the interval of the solution must be specified, or else I believe the space of solutions is infinite dimensional.