The DE is...

(1)

... and we suppose its solutions are expandable is McLaurin series as...

(2)

From (2) we derive first...

(3)

The (1) can be solved only if we know the 'initial conditions' that usuallly are expressed as , . If now we impose these 'initial conditions' we obtain...

Now we can valuate the succesive terms inserting (2) and (3) in (1)...

... and so on...

Kind regards