You are right, there is a typo in my post.
But
eq = s^4 y + 19/10 s^3 y + 487/30 s^2 y + 69 s y + 45/10 y == 5 va[8];
yy = y /. Solve[eq, y][[1, 1]]
( * (150 va[8])/(135 + 2070 s + 487 s^2 + 57 s^3 + 30 s^4) *)
this gives the same peculiar Plot ( perhaps Neil means this as unstable behaviour) as before, even with 57 as coefficient of s^3.
There is another remark. You are looking for a differential equation in the t-domain which gives your equation in the s-domain. Independent of the initial values of f what do you say to this (but I don't have any idea how to handle such an equation)
dgl = 9/2 y DiracDelta[t] + 69 DiracDelta[t] f[0] -
5 DiracDelta[t] va[8] + 487/30 f[0]
\!\(\*SuperscriptBox["DiracDelta", "\[Prime]",
MultilineFunction->None]\)[t] + 487/30 DiracDelta[t]
\!\(\*SuperscriptBox["f", "\[Prime]",
MultilineFunction->None]\)[0] + 19/10
\!\(\*SuperscriptBox["DiracDelta", "\[Prime]",
MultilineFunction->None]\)[t]
\!\(\*SuperscriptBox["f", "\[Prime]",
MultilineFunction->None]\)[0] + 69
\!\(\*SuperscriptBox["f", "\[Prime]",
MultilineFunction->None]\)[t] + 19/10 f[0]
\!\(\*SuperscriptBox["DiracDelta", "\[Prime]\[Prime]",
MultilineFunction->None]\)[t] +
\!\(\*SuperscriptBox["f", "\[Prime]",
MultilineFunction->None]\)[0]
\!\(\*SuperscriptBox["DiracDelta", "\[Prime]\[Prime]",
MultilineFunction->None]\)[t] + 19/10 DiracDelta[t]
\!\(\*SuperscriptBox["f", "\[Prime]\[Prime]",
MultilineFunction->None]\)[0] +
\!\(\*SuperscriptBox["DiracDelta", "\[Prime]",
MultilineFunction->None]\)[t]
\!\(\*SuperscriptBox["f", "\[Prime]\[Prime]",
MultilineFunction->None]\)[0] + (487
\!\(\*SuperscriptBox["f", "\[Prime]\[Prime]",
MultilineFunction->None]\)[t])/30 + f[0]
\!\(\*SuperscriptBox["DiracDelta",
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[t] + DiracDelta[t]
\!\(\*SuperscriptBox["f",
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[0] + 19/10
\!\(\*SuperscriptBox["f",
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[t] +
\!\(\*SuperscriptBox["f",
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None]\)[t]
(That looks somewhat weird, just copy and paste it, it can for sure be better written))
Then the LaplaceTransform is essentially your equation ( with the 5 va[8] term on the left side)
In[17]:= eq4 = Expand[LaplaceTransform[dgl, t, s] /. LaplaceTransform[f[t], t, s] -> y]
Out[17]= (9 y)/2 + 69 s y + (487 s^2 y)/30 + (19 s^3 y)/10 + s^4 y - 5 va[8]