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Math Help - help with differential equations

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    8

    help with differential equations

    Hi there , i have the following problam , and i sort of got stuck in the middle
    anyways i have a system of 2 differential equations so here it is

    y'=y-10z
    z'=y-z+3cos(2t)

    seems fairly simple , anyways i reached a point where i have Z(s) and Y(s) via laplace transfrom and well , got a bit stuck with it. so to save you time

    Y(s) = (s^3 + S^2 -26s +4)/((s^2 + 4)(s^2 + 9))
    Z(s) = (2s^2 - 3s +4)/((s^2 +4)(s^2 + 9))

    i am fairly sure of the Y(s) and Z(s) i've recieved so anyone kind enough to help , might as well skip the middle part of reaching those two. anyways my request is a way to simplify Y(s) and Z(s) to the point where i can do L^-1[Y(s)] and L^-1[Z(s)] and ofc the way it was done. help would be much appriciated.
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  2. #2
    Super Member
    Joined
    Aug 2008
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    Partial fractions right?

    \frac{g(s)}{(s^2+4)(s^2+9)}=\frac{as+b}{s^2+4}+\fr  ac{cs+d}{s^2+9}

    Or just ask Mathematica:

    Code:
    In[13]:=
    LaplaceTransform[InverseLaplaceTransform[
       (s^3 + s^2 - 26*s + 4)/((s^2 + 4)*
         (s^2 + 9)), s, t], t, s]
    
    Out[13]=
    (1/3)*(-((18*s)/(4 + s^2)) + 3/(9 + s^2) + 
       (21*s)/(9 + s^2))
    . . . what, I ain't proud.
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  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    8
    Thanks Shawsend , been a while since i've done any partial fractions , wasnt sure if g(s) could be just about any function or if it was restricted somehow.
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  4. #4
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Degree of denominator > degree of numerator for partial fractions. Also, just learned could use \text{Apart} command in Mathematica for this:

    Code:
    In[17]:=
    Apart[(s^3 + s^2 - 26*s + 4)/
       ((s^2 + 4)*(s^2 + 9))]
    
    Out[17]=
    -((6*s)/(4 + s^2)) + (1 + 7*s)/(9 + s^2)
    Caveat: I do not wish to promote using Mathematica until you're comfortable with doing it yourself first.
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