find the power series solution about x=0 with y(0)=1 and show:
y is a polynomial iff lambda is a non-negative integer n.
We start supposing that the second order linear ODE...
, (1)
... has an analytic solution of the form...
(2)
If we substitute the expression (2) directly in (1) we obtain...
(3)
... which immediately gives us...
(4)
If we set the (4) allow us to derive all the . It is evident from (4) that if with an integer, all the with vanish, so that is a polynomial of degree ...
Kind regards