I tried to solve the product below: 3 t sin ⁡ ( 6 t )but it seems that getting the transform of each and multiply the result is not leading to a correct answer: 3 s 2 6 s 2 + 36 How does one solve such transforms?

Question

I tried to solve the product below:  3  t  sin  ⁡  (  6  t  )but it seems that getting the transform of each and multiply the result is not leading to a correct answer:      3          s      2            6                  s        2            +      36      How does one solve such transforms?

poly3E7V · Accepted Answer

We know that if   L  (  f  (  t  )  )  =  F  (  s  ) so   L  (  t  .  f  (  t  )  )  =  −      F    ′    (  s  ) in which       F    ′    (  s  )  =            d      F              d      s      . Here you need just to derivative the second part of the last formula above with respect to s and then multiply the result by 3

Will Osborn · Accepted Answer

In fact, if you want to know       L        {          t      n        f    (    t    )    }  , you can use this formula (which can be proven using integration by parts):      L        {          t      n        f    (    t    )    }    =  (  −  1      )    n              d      n              d              s        n                  L        {    f    (    t    )    }  So it kind of turn polynomials multiplying into derivatives.

Solve the product below: 3t sin(6t)

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