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Math Help - Partial Fractions

  1. #1
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    Partial Fractions

    I am in the middle of solving a integro-differential, and I need to use inverse Laplace transforms to complete the last bit, I have the following:

     <br />
x(s)= \frac{1}{s\left[1+\frac{s}{s+1}\right]}+ \frac{1}{s^2\left[1+\frac{s}{s+1}\right]}-\frac{1}{s+1\left[1+\frac{s}{s+1}\right]}<br />

    I have simplified this to the following form:

     <br />
x(s)= \frac{s+1}{s(s-2)}+\frac{s+1}{s^2(2s+1)}-\frac{s+1}{(s+1)^2+s}<br />

    Then using partial fractions I have taken the above term by term, and done the following:

     <br />
\frac{s+1}{s(s-2)} = \frac{-1/2}{s}+\frac{3/2}{s-2}<br />

    Second term:

     <br />
\frac{s+1}{s^2(2s+1)} = \frac{-1}{s}+\frac{1}{s^2}+\frac{2}{2s+1}<br />

    Can some please varify if I have used partial fractions correctly on the first 2 expressions

    Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?
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  2. #2
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    Quote Originally Posted by ubhik View Post
    I am in the middle of solving a integro-differential, and I need to use inverse Laplace transforms to complete the last bit, I have the following:

     <br />
x(s)= \frac{1}{s\left[1+\frac{s}{s+1}\right]}+ \frac{1}{s^2\left[1+\frac{s}{s+1}\right]}-\frac{1}{s+1\left[1+\frac{s}{s+1}\right]}<br />

    I have simplified this to the following form:

     <br />
x(s)= \frac{s+1}{s(s-2)}+\frac{s+1}{s^2(2s+1)}-\frac{s+1}{(s+1)^2+s}<br />

    Then using partial fractions I have taken the above term by term, and done the following:

     <br />
\frac{s+1}{s(s-2)} = \frac{-1/2}{s}+\frac{3/2}{s-2}<br />

    Second term:

     <br />
\frac{s+1}{s^2(2s+1)} = \frac{-1}{s}+\frac{1}{s^2}+\frac{2}{2s+1}<br />

    Can some please varify if I have used partial fractions correctly on the first 2 expressions

    Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?
    You have.

    (s+1)^2+s to get s^2 + 3s + 1. This factorises, but not nicely.
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  3. #3
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    Quote Originally Posted by ubhik View Post
     <br />
x(s)= \frac{s+1}{s(s-2)}+\frac{s+1}{s^2(2s+1)}-\frac{s+1}{(s+1)^2+s}<br />

    Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?
    Break it in a different way:


    \frac{s+1}{s^2+3s+1} = \frac{s+1}{(s+\frac32)^2 - \frac54}
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