How to compute the inverse Laplace transformation of s 2 ( s 2 + 1 ) 2

Name: How to compute the inverse Laplace transformation of s 2 ( s 2 + 1 ) 2
Uploaded: 2022-02-23T17:22:57+00:00
Description: How to compute the inverse Laplace transformation of s 2 ( s 2 + 1 ) 2

Question

How to compute the inverse Laplace transformation of             s      2              (              s        2            +      1              )        2

Illuddybopylu · Accepted Answer

F  (  s  )  =            s      2              (              s        2            +      1              )        2              =                    s        2            +      1              (              s        2            +      1              )        2              −      1          (              s        2            +      1              )        2              F  (  s  )  =      1                  s        2            +      1        +            1              2        s                        d              d        s                  1          (              s        2            +      1      )        .Apply inverse Laplace Transform:  f  (  t  )  =  sin  ⁡  t  −            1      2            ∫    0    t    1  ×  τ  sin  ⁡  τ     d  τ  f  (  t  )  =            1      2        (  sin  ⁡  t  +  t  cos  ⁡  t  )

Skye Vazquez · Accepted Answer

Consider the Laplace transforms of   sin  ⁡  (  a  t  ) and   t    cos  ⁡  (  a  t  ) which are                    sin        ⁡        (        a        t        )                            ≑                  a                                    s              2                        +                          a              2                                      =                              a                                      s              2                        +                          a              3                                            (                          s              2                        +                          a              2                                      )                              2                                                                        t                cos        ⁡        (        a        t        )                            ≑                                            s              2                        −                          a              2                                            (                          s              2                        +                          a              2                                      )                              2                                                    .            Now one can notice that            f      ¯        (  s  )              g      ¯        (  s  )  ≑      ∫          0              t        f  (  t  −  u  )    g  (  u  )    d  uA second method is to use the convolution theorem,f¯(s)g¯(s)≑∫t0f(t−u)g(u)duwhere             f      ¯        (  s  ) is the transform of   f  (  t  ). Since  cos  ⁡  (  a  t  )  ≑      s                  s        2            +              a        2            then                                          s            2                                (                          s              2                        +                          a              2                                      )              2                                                          ≑                  ∫                      0                                t                          cos        ⁡        (        a        u        )                cos        ⁡        (        a        t        −        a        u        )                d        u                                          ≑                              [                                          2                a                t                                cos                ⁡                (                a                t                )                −                sin                ⁡                (                a                (                t                −                2                u                )                )                                            4                a                                      ]                                0                                t                                                            ≑                  1                      2            a                                            (          sin          ⁡          (          a          t          )          +          a                    t                    cos          ⁡          (          a          t          )          )                .            Modified form             s      3              (              s        2            +      1              )        2            By using :                    cos        ⁡        (        a        t        )        −        a        t                sin        ⁡        (        a        t        )                            ≑                              s                        (                          s              2                        −                          a              2                        )                                (                          s              2                        +                          a              2                                      )              2                                                          cos        ⁡        (        a        t        )        +        a        t                sin        ⁡        (        a        t        )                            ≑                              s                        (                          s              2                        +            3                          a              2                        )                                (                          s              2                        +                          a              2                                      )              2                                          then it can be shown that      1    2      (  2    cos  ⁡  (  a  t  )  −  a    t    sin  ⁡  (  a  t  )  )  ≑            s      3              (              s        2            +              a        2                    )        2              .

How to compute the inverse Laplace transformation of (s^2)/((s^2+1)^2)

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