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Math Help - Getting stuck on a simple inverse Laplace transform. How do I massage this fraction?

  1. #1
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    Getting stuck on a simple inverse Laplace transform. How do I massage this fraction?

    I'm having some difficulty with what should be a simple (one sided) inverse Laplace transform:

    F(s) = \frac{2s}{s + 6}


    Now, I know the answer is 2 \delta (t) - 12 e^{-6t} u(t)

    I'm just not sure how to get there. I'm not looking to apply the definition - I'm looking to massage the fraction into standard transform pairs or use standard properties of the transform.


    It seems to me that I need to accomplish this:

    \frac{2s}{s + 6} = 2 - \frac{12}{s + 6}


    I'll probably kick myself for this, but what am I missing?
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  2. #2
    Senior Member BAdhi's Avatar
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    Re: Getting stuck on a simple inverse Laplace transform. How do I massage this fract

    you can get it like this easily,

    \frac{2s}{s+6}=\frac{2(s+6)-12}{s+6}=\frac{2(s+6)}{s+6}-\frac{12}{s+6}=2-\frac{12}{s+6}

    then remember that

    L[\delta(t)]=1 and L[u(t)]=\frac{1}{s} and  L[e^{at}f(t)]=F(s-a) where L[f(t)]=F(s)
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  3. #3
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    Re: Getting stuck on a simple inverse Laplace transform. How do I massage this fract

    Or "long division". s divides into 2s twice. 2 times s+ 6 is 2s+ 12 and subtracting: (2s+ 0)- (2s+ 12)= -12. s+ 6 divides into 2s twice with remainder -12:
    \frac{2s}{s+ 6}= 2+ \frac{-12}{s+ 6}.
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  4. #4
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    Re: Getting stuck on a simple inverse Laplace transform. How do I massage this fract

    Quote Originally Posted by BAdhi View Post

    you can get it like this easily,

    \frac{2s}{s+6}=\frac{2(s+6)-12}{s+6}=\frac{2(s+6)}{s+6}-\frac{12}{s+6}=2-\frac{12}{s+6}

    Yup, I was right. Kicking myself. This part of Laplace transforms is an art. Dammit.

    Thanks for showing me what I was missing.
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