Solve $y''+2ty'+4y=0$. 
Using power series method (or any other method), ﬁnd a polynomial solution to the
  following problem
$$y''+2ty'+4y=0$$

I am not sure how to solve this question. Using the power series method I always end up with a complicated relation  $a_n= -2\frac{a_{n-1}(3-n)}{n(n+1)}$, and I'm not sure if this is the correct way to continue.
 A: I hate to be the bearer of evil tidings, but the equation
$y'' + 2ty' + 4y = 0 \tag{1}$
has no non-trivial polynomial solutions.  For suppose
$p(t) = \sum_0^n p_j t^j  \ne 0 \tag{2}$
were a solution of (1); note that $\deg p = n$, that is, we assume $p_n \ne 0$.  Suppose $n \ge 2$; then since $\deg p'' = n - 2$, we see that the degree $n$ term of $p''(t) + 2tp'(t) + 4p(t)$ is $2np_n t^n + 4p_n t^n$ and thus, since $p(t)$ solves (1), 
$2np_n t^n + 4p_n t^n = 0, \tag{3}$
or, since $p_n t^n \ne 0$,
$2n + 4 = 0.  \tag{4}$
But the solution to (4) is
$n = -2, \tag{5}$
a contradiction to the assumption $p(t)$ is a polynomial of degree $n \ge 2$.  The remaining option is $n \le 1$;   but if $p(t) = p_1 t + p_0$,
then (1) yields
$2tp_1 + 4p_1t + 4p_0 = 6p_1t + 4p_0 = 0; \tag{5}$
but (5) forces $p_1 = p_0 = 0$  Thus only trivial polynomials satisfy (1).
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
A: My result (below) is almost the same as the formula given by user64494 from Maple (different typography) and knowing that $-i$ erf($i t$) = erfi($t$)

Expressed on the form of series, the solutions are :

A: The Maple code 
dsolve(((D@@2)(y))(t)+2*t*(D(y))(t)+4*y(t) = 0)

outputs 
$$ y(t)=\_C1e^{-t^2}t+\_C2i \operatorname{erf}(it)e^{-t^2}\sqrt{\pi}t+1)$$ 
and 
dsolve(((D@@2)(y))(t)+2*t*(D(y))(t)+4*y(t) = 0, y(t), series, order = 9)

produces 
$$y \left( t \right) =(y \left( 0 \right) +\mbox {D} \left( y \right) 
 \left( 0 \right) t-2\,y \left( 0 \right) {t}^{2}-\mbox {D} \left( y
 \right)  \left( 0 \right) {t}^{3}+{\frac {4\,y \left( 0 \right) }{3}}
{t}^{4}+{\frac {\mbox {D} \left( y \right)  \left( 0 \right) }{2}}{t}^
{5}-{\frac {8\,y \left( 0 \right) }{15}}{t}^{6}-{\frac {\mbox {D}
 \left( y \right)  \left( 0 \right) }{6}}{t}^{7}+{\frac {16\,y \left( 0
 \right) }{105}}{t}^{8}+O \left( {t}^{9} \right) ).
 $$
