See the attachment for the details but the x doesn't magically disappear

but the x term is incorporated into determining the coefficient of the x term once the power series and its derivative are plugged back into the differential equation

You'll notice c1-c0 = 0 considering the contstant term

then 2c2-c1+1 = 0 in considering the linear term this is where x comes in

There are no singularities in this DE