Laplace Transforms

Here are some examples relating to some problems from the early part of the course. Some of the results below have application to the damped oscillator problem.

<<Calculus`LaplaceTransform`

LaplaceTransform[Cosh[t], t, s]

   s
-------
      2
-1 + s

LaplaceTransform[Sinh[t], t, s]

   1
-------
      2
-1 + s

InverseLaplaceTransform[1/(k+s), s, t]

 -(k t)
E

InverseLaplaceTransform[1/(s), s, t]

1

InverseLaplaceTransform[1/(s(s+k)), s, t]

1     1
- - ------
k    k t
    E    k

InverseLaplaceTransform[1/(s^2 + eom*s + kom), s, t]

                         2
   -(eom t)/2 - (Sqrt[eom  - 4 kom] t)/2
  E
-(--------------------------------------) + 
                    2
            Sqrt[eom  - 4 kom]
 
                         2
   -(eom t)/2 + (Sqrt[eom  - 4 kom] t)/2
  E
  --------------------------------------
                    2
            Sqrt[eom  - 4 kom]

InverseLaplaceTransform[s/(s^2 + eom*s + kom), s, t]

                    2
   ((-eom + Sqrt[eom  - 4 kom]) t)/2                2
-(E                                  (eom - Sqrt[eom  - 4 kom]))
---------------------------------------------------------------- + 
                                2
                      2 Sqrt[eom  - 4 kom]
 
                               2
                 eom + Sqrt[eom  - 4 kom]
  ------------------------------------------------------
                     2
     ((eom + Sqrt[eom  - 4 kom]) t)/2         2
  2 E                                 Sqrt[eom  - 4 kom]

InverseLaplaceTransform[1/(s+1)(s-1), s, t]

-2
-- + DiracDelta[t]
 t
E