Hakan,
Let me second Michael again: I love the forum and the exchanges it stimulates!
Now, I looked again at the code I posted at 4:30 am my time (EST), and I did mess up copying parts of it (In[15] and Out[15]), NOT applying the initial condtions. I have now corrected these two lines in that post. Here is my code again, this time carefully run and copied:
In[1] := eqn = a t^2 y''[t] + b y'[t] + c y[t] == 0;
In[2] := leqn = LaplaceTransform[eqn, t, s] /. {y[0] -> 0, y'[0] -> 1}
Out[2] := c LaplaceTransform[y[t], t, s] +b s LaplaceTransform[y[t], t, s]
+a LaplaceTransform[t^2 y'' [t], t, s] == 0
In[3] := sol = SolveValues[leqn, LaplaceTransform[y[t], t, s]]
Out[3] := {-a LaplaceTransform[t^2 y'' [t], t, s])/(c + b s)}
Note that the term LaplaceTransform[t^2 y'' [t], t, s] is unresolved. Mathematica doesn't know what to do with it. You missed this part again by not printing out the output between your lines starting with lt2 and sol. I'm using Mathematica 12.3 right now. But I doubt that version 13 can resolve Laplace transform of a product t^2 y'' [t]. Here is what I get:
In[4] := LaplaceTransform[t^2 y''[t], t, s]
Out[4] := LaplaceTransform[t^2 y''[t], t, s]
Which means: "I can't do it!" Please let us know what you get!
Again, thanks, Hakan, for finding this fascinating problem. I can't wait for Luke to weigh in ...
Best,
Zbigniew