Cheat sheet(s) for trial particular solutions of 2nd order homogenous differential equations $~\text{e.g}~~\frac{d^{2}y}{dt^2}+2t\frac{dy}{dt}+ty=0~$ $$   \frac{d^{2}y}{dt^2}+ 2t \frac{dy}{dt}+ t y=0 ~~  \tag{1}   $$
At least I know that in this case of ODE can be solved by finding out 2 particular solutions.
As those 2 particular solutions are known, the general solution for this ODE can be written as below form.
$$  y\left(x\right)=C_{1}y_{1}\left(x\right)+ C_{2} y_{2}\left(x\right)  $$
I've been completely struggling to find it.
Is there some cheat sheet(s) which can be used to find out particular solutions so that I don't have to ask about it in MSE?
Or can you give me some hint(s) so that I can find the particulr solutions?
I've found of it one(at 13th page) (however not applicable to this ODE though).
I assume computer cannot be used and this problem is solved in a offline paper test.
The wolfram showed the below result but what are Ai and Bi ?

 A: You can ask for series solutions. In this case proposing $y_n = \sum_{k=0}^n a_k t^k$ and substituting into the ODE we have
$$
\left(\sum_{k=0}^n a_k t^k\right)''+2t\left(\sum_{k=0}^n a_k t^k\right)'+t\sum_{k=0}^n a_k t^k=0
$$
grouping the powers of $t$ we have
$$
2a_2+(a_0+2a_1+6a_3)t+(a_1+4a_2+12a_4)t^2+\cdots+ = 0
$$
to maintain the nullity, the $t^k$ coefficients should be null. The first condition thus is $a_2 = 0$. After that, as we need two initial independent conditions, we will intent to solve to nullity all the coefficients depending on $a_0, a_1$. Thus, after making $a_2=0$ we have
$$
\cases{
a_0+2 a_1+6 a_3 = 0\\
a_1+12 a_4= 0\\
6 a_3+20 a_5=0\\
a_3+8 a_4+30 a_6=0\\
a_4+10 a_5+42 a_7=0\\
\vdots
}
$$
and solving for $a_0,a_1$ we have
$$
\cases{
 a_3=-\frac{1}{6} \left(a_0+2 a_1\right) \\
 a_4=-\frac{a_1}{12} \\
 a_5=\frac{1}{20} \left(a_0+2 a_1\right) \\
 a_6=\frac{1}{180} \left(a_0+6 a_1\right) \\
\vdots
}
$$
thus we have a whole solution.
A: Maple:
$$y \left( t \right) ={\it \_C1}\,{{\rm e}^{-t/2}}{{ \rm KummerM}\left(1/16,\,1
/2,\,-1/4\, \left( 2\,t-1 \right) ^{2}\right)}+{\it \_C2}\,{{\rm e}^{-
t/2}}{{\rm KummerU}\left(1/16,\,1/2,\,-1/4\, \left( 2\,t-1 \right) ^{2}
\right)}
$$
