Avoid problem with solving the LaplaceTransform and inversing it?

Questions
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1

Hello all,

I have been working on a project involving the Laplace equation.
Sadly enough, somehow I don’t get it to work.
I have tried looking for a solution in the manual of Mathematica and tried several solutions, but non of them seem to work.
Every time I end up getting the same thing, with the words LaplaceTransform and InverseLaplaceTransform, as I put in.
I have tried using different domains for the NSolve, as well as reformulating the whole function which needs to be solved and trying different ways of formulating the variable t in the NSolve.
The idea is that I get the Laplace transformation of the function (2nd line in the code), get a solution for variable t. After which the inverse Laplace transformation should be taking from this solution.
My starting code is as following:

Clear["Global`*"]
Sin[\[Omega] t] == 
 q''[t]/Subscript[dw, 0]^2 + (2 \[Xi]q'[t])/Subscript[dw, 0] + q[t]/d

LaplaceTransform[%, t, s]
NSolve[%, t]
InverseLaplaceTransform[%, s, t]

Gratitude in advance.

Kind regards,
Justin Warners

2

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3

Hello

You made a syntax mistake.

 Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2 \[Xi]q'[t])/Subscript[dw, 0] + q[t]/d

It should be:

 Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2 \[Xi] q'[t])/Subscript[dw, 0] + q[t]/d

You forgot a backspace to add between \xi and q’(t)

NSolve is a numerical solver, can’t solve a symbolically.

   Clear["Global`*"]

   eq = Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2*\[Xi] *q'[t])/Subscript[dw, 0] + q[t]/d

   eq1 = LaplaceTransform[eq, t, s] /. q[0] -> 0 /. q'[0] -> 0 (* Initial conditions,you may delete it or change to:  eq1 = LaplaceTransform[eq, t, s] *)

   sol = First@Solve[eq1, LaplaceTransform[q[t], t, s]] // Simplify

   q[t] = InverseLaplaceTransform[LaplaceTransform[q[t], t, s] /. sol, s, t]
4

First, I think, you have a typo. There should be a blank between [Xi] and q’[t]).

Then as far as I know one should solve the transformed equation for the Laplacetransform and then apply the inverse transformation (if possible).
I did that as outlined in the notebook attached