Find a series solution to the following initial value problem:
$x''(t) + 2tx'(t) − 8x(t) = 0 ∀t > 0$ subject to $x(0) = 1$ and $x'(0) = 0$
Ive got that $x(t) = \sum\limits_{n=0}^\infty a_{n}t^{n}$,
$x'(t) = \sum\limits_{n=1}^\infty na_{n}t^{n-1}$
and $x''(t) = \sum\limits_{n=2}^\infty (n-1)na_{n}t^{n-2}$
But I need some guidance through this question as I seem to always go wrong somewhere.
Any help will be appreciated.