Inverse Laplace Transform Examples

In[27]:=

  InverseLaplaceTransform[4/(s-a),s,t]

Out[27]=

     a t
  4 E

In[28]:=

  InverseLaplaceTransform[1/((s-a)(s-b)),s,t]

Out[28]=

   a t     b t
  E       E
  ----- - -----
  a - b   a - b

In[29]:=

  InverseLaplaceTransform[1/(s-a)^2,s,t]

Out[29]=

   a t
  E    t

In[30]:=

  InverseLaplaceTransform[1/(s-a)^6,s,t]

Out[30]=

   a t  5
  E    t
  -------
    120

In[31]:=

  InverseLaplaceTransform[6 s/(s^2+4 s+13),s,t]

Out[31]=

     Cos[3 t]   2 Sin[3 t]
  6 (-------- - ----------)
        2 t          2 t
       E          3 E

In[32]:=

  InverseLaplaceTransform[s^2/(s^2+6 s-6),s,t]

Out[32]=

      -3 t - Sqrt[15] t          3   -3 t - Sqrt[15] t
  -3 E                  - 4 Sqrt[-] E                  - 
                                 5
   
       -3 t + Sqrt[15] t          3   -3 t + Sqrt[15] t
    3 E                  + 4 Sqrt[-] E                  + DiracDelta[t]
                                  5

Cool! Remember, Mathematica knows the Dirac Delta function!

In[33]:=

  InverseLaplaceTransform[1,s,t]

Out[33]=

  DiracDelta[t]

Aside on the Dirac Delta Function