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Math Help - Laplacing ambigious equation, wants explanation:)

  1. #1
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    Laplacing ambigious equation, wants explanation:)

    Hello there, im trying out this from a workbook,I have no one to explain to me.

    Could someone tell me the proper procedure of of laplacing this? it seems it is using the function of multiplication t, which is -F'(s) , but not as near to the answer(attached), could someone explain to me? especially the extra +(5/(s-1)) value which i dont think it comes from partial fraction. Thank you, kamsahamnida.
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  2. #2
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    Quote Originally Posted by kolowee9 View Post
    Hello there, im trying out this from a workbook,I have no one to explain to me.

    Could someone tell me the proper procedure of of laplacing this? it seems it is using the function of multiplication t, which is -F'(s) , but not as near to the answer(attached), could someone explain to me? especially the extra +(5/(s-1)) value which i dont think it comes from partial fraction. Thank you, kamsahamnida.
    There are several ways of doing this, including the following:

    LT\left[ t e^{t - 5} u(t - 5)\right] = \int_5^\infty t e^{t - 5} e^{-st} \, dt

    Make the substitution u = t - 5:

    = \int_0^\infty (u + 5) e^{u} e^{-s(u + 5)} \, dt = e^{-5s}\int_0^\infty (u + 5) e^{u} e^{-su} \, dt

    = e^{-5s} \left( \int_0^\infty ue^{u} e^{-su} \, dt + 5 \int_0^\infty e^{u} e^{-su} \, dt \right)

    and the rest should be clear.


    Spoiler:
    In fact, since this is essentially just e^{-5s} \left( LT[x e^x]  + 5 LT[e^x] \right) you can just use tables and a Shifting Theorem to finish it off.
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