How to solve the transforms below step by step L − 1 1 ( s + λ ) 2 − ω 2 L − 1 a ( s + 2 λ ) + b ( s + λ ) 2 − ω 2 I found some transformed tables, but I couldn't understand where the result comes from, or how to reproduce the accounts step by step.

Question

How to solve the transforms below step by step            L              −      1            1          (      s      +      λ              )        2            −              ω        2                        L              −      1                  a      (      s      +      2      λ      )      +      b              (      s      +      λ              )        2            −              ω        2            I found some transformed tables, but I couldn&#039;t understand where the result comes from, or how to reproduce the accounts step by step.

allelog5 · Accepted Answer

These come from the Laplace transforms of       e          −      λ      t        cosh  ⁡  (  ω  t  ) and       e          −      λ      t        sinh  ⁡  (  ω  t  ):      L        [          e              −        λ        t              sinh    ⁡    (    ω    t    )    ]    =      1    2        (          L              [              e                  −          (          λ          −          ω          )          t                    ]        −          L              [              e                  −          (          λ          +          ω          )          t                    ]        )    =      1    2        (          1              s        +        λ        −        ω              −          1              s        +        λ        +        ω              )    =      ω          (      s      +      λ              )        2            −              ω        2                  L        [          e              −        λ        t              cosh    ⁡    (    ω    t    )    ]    =      1    2        (          L              [              e                  −          (          λ          −          ω          )          t                    ]        +          L              [              e                  −          (          λ          +          ω          )          t                    ]        )    =      1    2        (          1              s        +        λ        −        ω              +          1              s        +        λ        +        ω              )    =            s      +      λ              (      s      +      λ              )        2            −              ω        2            To solve those Laplace transforms, you simply need to write them as linear combinations of the above two transforms.

How to solve the transforms below step by step ccL^(-1) (1)/((s+lambda)^2-omega^2) ccL^(-1) (a(s+2lambda)+b)/((s+lambda)^2-omega^2)

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