Let's write the differential equation as...

(1)

Now we suppose that (1) has a solution y(*) that can be expressed as power sum, so that is...

(2)

In this case the identity (1) becomes [the intermediate computations are not reported...] the following...

(3)

Now we try to compute the from (3) imposing that the coefficients ot the terms are identical in both terms. Any solution of (1) is defined unless an arbitrary constant, so that we are allowed to set...

Equating in (3) the coefficients of the term we obtain...

... all right!... equating the the coefficients of the term we obtain...

... all right!... equating the the coefficients of the term we obtain...

... ehmm!... there is some minor difficult in computing ...

... the conclusion is that our hypothesis is false and doesn't exist any solution of (1) that can be expressed in power series like (2) ...

Kind regards