I have recently found a very promising approach [1] to tackling those questions and I thought I would present it in here. Firstly let me describe the algorithm itself and then apply it to our problem and find a whole class of closed form solutions for some functional form of $\omega(t)$.
I will change the notation slightly just to be inline with that in the paper.
The idea is to seek for solutions our ${\bf input}$ equation:
\begin{equation}
y^{''}(x) = v(x)\cdot y(x)
\end{equation}
in the following form $y_{1,2}(x) = m(x) \cdot F_{1,2}(\xi(x))$ (ansatz) where $m(x)$ and $\xi(x)$ are functions to be specified later and $F_{1,2}(\cdot)$ are fundamental solutions to some other second order ODE to be termed the ${\bf target}$ equation. In other words we have:
\begin{equation}
\left(\frac{d^2}{d x^2} + a_1(x) \frac{d}{d x} + a_0(x)\right) F_{1,2}(x)=0
\end{equation}
Now by inserting the ansatz into the input equation and equating the coefficients at $F$ and at $F^{'}$ to zero we end up with following equations for the functions in question. We have:
\begin{eqnarray}
\frac{m^{'}}{m}&=& \frac{1}{2}\left(
a_1 \cdot \xi^{'} -
\frac{\xi^{''}}{\xi^{'}}
\right)\\
0&=&\left( a_1^2+ 2 a_1^{'} - 4 a_0\right) \cdot [\xi^{'}]^4 + 3 [\xi^{''}]^2-2 \xi^{'''} \cdot \xi^{'} - 4 v \cdot [\xi^{'}]^2
\end{eqnarray}
The idea is now to search for solutions to the bottom equation above in form of rational functions.
By matching the degree of that rational function (difference in degrees of the numerator and denominator) one can deduce the form of the denominator and also the degree of the numerator and then by inserting those into the equation get some non-linear equations for parameters , equations that are easily solved. I refer the reader to the paper below for details. This method seems to be quite powerful provided we properly chose the target equation. It is conceivable that the target equation should be related to the hypergeometric equation.
Now, to demonstrate the power of this method let us take $v(x) = a \cdot x^{2 n} + b \cdot x^{n-1}$ where $n\ne-1$. Then it appears that the fundamental solutions to our input equation read:
\begin{eqnarray}
\left\{
\frac{1}{\sqrt{x^n}} M_{\frac{-b}{(2n+2) \sqrt{a}},\frac{1}{2n+2}}\left( \frac{2}{n+1} \sqrt{a} x^{n+1}\right),
\frac{1}{\sqrt{x^n}} W_{\frac{-b}{(2n+2) \sqrt{a}},\frac{1}{2n+2}}\left( \frac{2}{n+1} \sqrt{a} x^{n+1}\right)
\right\}
\end{eqnarray}
where $M$ and $W$ are the Whittaker functions (which in turn are related to the confluent hypergeometric equation). Below is a quick and dirty proof of this result
In[648]:= a =.; b =.; n =.; x =.;
FullSimplify[(D[#, {x, 2}] - (a x^(2 n) + b x^(n - 1)) #) & /@ {1/
Sqrt[x^n] WhittakerM[-b/((2 n + 2) Sqrt[a]), 1/(2 n + 2),
2/(n + 1) Sqrt[a] x^(n + 1)],
1/Sqrt[x^n] WhittakerW[-b/((2 n + 2) Sqrt[a]), 1/(2 n + 2),
2/(n + 1) Sqrt[a] x^(n + 1)]}]
Out[649]= {0, 0}
Update: I have been playing around with this algorithm and to my great surprise I realized that the solution given above is a special case of another broad class of solutions below. Let us take $v(x) = a \cdot x^{2 n} + b \cdot x^{n-1} + c \cdot x^{-2}$ where $n \ne -1$. Then the fundamental solutions to our input equation read:
\begin{eqnarray}
\left\{
\frac{1}{\sqrt{x^n}} M_{\frac{-b}{(2n+2) \sqrt{a}},\frac{\sqrt{1+4 c}}{2n+2}}\left( \frac{2}{n+1} \sqrt{a} x^{n+1}\right),
\frac{1}{\sqrt{x^n}} W_{\frac{-b}{(2n+2) \sqrt{a}},\frac{\sqrt{1+4 c}}{2n+2}}\left( \frac{2}{n+1} \sqrt{a} x^{n+1}\right)
\right\}
\end{eqnarray}
Again, the line below "proves" this result. We have:
In[1249]:= a =.; b =.; c =.; x =.; n =.;
FullSimplify[(D[#, {x,
2}] - (a x^(2 n) + b x^(n - 1) + c x^-2) #) & /@ {1/Sqrt[x^n]
WhittakerM[-(b/((2 n + 2) Sqrt[a])), Sqrt[
1 + 4 c]/(2 n + 2) , (2 Sqrt[a])/(n + 1) x^(n + 1)],
1/Sqrt[x^n]
WhittakerW[-(b/((2 n + 2) Sqrt[a])), Sqrt[
1 + 4 c]/(2 n + 2) , (2 Sqrt[a])/(n + 1) x^(n + 1)]}]
Out[1250]= {0, 0}
Having said all this it would be interesting to know if this result is known and if yes by what means has it been derived.
[1] M. Bronstein & S. Lafaille, ``Solutions of linear ordinary differential equations in terms of special functions'', Proceedings of ISSAC'2002, Lille, ACM Press, 23-28. https://www-sop.inria.fr/cafe/Manuel.Bronstein/
