March 29, 2017

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Use the fact that L{f'}=sF-f(0) to derive the formula for L{cosh at}

  • Calculus HELP! - ,

    cosh(at) = d/dt (1/a sinh(at))
    sinh(at) = d/dt (1/a cosh(at))

    Since L{f(at)} = 1/|a| F(s/a)
    L{cosh(at)} = sL{1/a sinh(t)}
    = s/a L{sinh(t)}

    L{sinh(at)} = sL{1/a cosh(at)}
    = s/a L{cosh(at)} - s/a


    L{cosh(at)} = s/a (s/a L{cosh(at)} - s/a
    L{cosh(at)}(1-s^2/a^2) = -s/a
    = (s/a^2)/(s^2/a^2 - 1)
    = s/(s^2-a^2)

    Better check my algebra in the middle steps. I think I dropped an "a" somewhere, but the final answer is correct. You work this in the same way you do integration by parts twice for trig functions.

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