Using partial fractions, find the inverse Laplace transform of the function f ( s ) = s 2 + 2 s 2 ( s 2 + 3 )

Question

Using partial fractions, find the inverse Laplace transform of the function  f  (  s  )  =                    s        2            +      2                      s        2            (              s        2            +      3      )

Dalia Cordova · Accepted Answer

s              2                        +            2                                              s              2                        (                          s              2                        +            3            )                                              =                  A          s                +                  B                      s            2                          +                  C                                    s              2                        +            3                                                            =                              A            s            (                          s              2                        +            3            )            +            B            (                          s              2                        +            3            )            +            C                          s              2                                                          s              2                        (                          s              2                        +            3            )                                                            =                              A                          s              3                        +            (            B            +            C            )                          s              2                        +            3            A            s            +            3            B                                              s              2                        (                          s              2                        +            3            )                              So equating the powers of s you have A=0, 3B=2 and B+C=1 that is B=2/3 and C=1/3 and then  f  (  s  )  =      2          3              s        2              +      1          3      (              s        2            +      3      )      So you have      L    {  f  (  s  )  }  =      [          2      3        t    +          1              3                  3                      sin    ⁡    (          3        t    )    ]    u  (  t  )

Using partial fractions, find the inverse Laplace transform of the function f(s)=(s^2+2)/(s^2(s^2+3))

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