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 May 4th, 2014, 09:18 PM #1 Newbie   Joined: May 2014 From: Australia Posts: 1 Thanks: 0 Inverse Laplace Transform Find the inverse Laplace transform of: (3s + s*exp(-4s) +2*exp(-4s) +s^3)/((s^2)(s-1)(s+1)) - Suggest using partial fractions (This is where I had trouble). Last edited by Bkalma; May 4th, 2014 at 09:25 PM.
 May 6th, 2014, 07:08 PM #2 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 626 Thanks: 90 Math Focus: Electrical Engineering Applications Hi Bkalma, and welcome to the forums. There may be a more elegant way to work this problem, but I took the simple approach of factoring out the $\displaystyle \large{e^{-4s}}$: $\displaystyle \large{\frac{3s+s\cdot e^{-4s}+2\cdot e^{-4s}+s^3}{s^2(s-1)(s+1)}=\underbrace{\frac{s^3+3s}{s^2(s-1)(s+1)}}_{\text{term 1}}+\underbrace{\frac{(s+2)}{s^2(s-1)(s+1)}}_{\text{term 2}}\cdot e^{-4s}}$ The idea, as you can probably tell, is to use partial fractions on term 1 and term 2. Then for term 2, use the Laplace Transform pair: $\displaystyle \large{f(t-t_0)\cdot u(t-t_0) \Leftrightarrow e^{-st_0}F(s)}$ to complete the solution. I will work through the partial fraction solution for term 1 and leave term 2 and the final solution for you to complete (but if you get stuck, feel free to reply). So for term 1: $\displaystyle \large{\frac{s^3+3s}{s^2(s-1)(s+1)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{s-1}+\frac{D}{s+1}}$ Multiply through by the denominator: $\displaystyle \large{s^3+3s=A(s)(s-1)(s+1)+B(s-1)(s+1)+Cs^2(s+1)+Ds^2(s-1)}$ [1] Letting s=0 in [1] we get: $\displaystyle \large{0=B(-1)(1) \rightarrow B=0}$ Letting s=1 in [1] we get: $\displaystyle \large{4=C(1)(2) \rightarrow C=2}$ Letting s=-1 in [1] we get: $\displaystyle \large{-4=D(1)(-2) \rightarrow D=2}$ Since every variable except for A is known, let's just plug in s=2 in [1] to find A: $\displaystyle \large{8+6=A(2)(1)(3)+0+2(2^2)(3)+2(2^2)(1) \rightarrow A=-3}$ So: $\displaystyle \large{\frac{s^3+3s}{s^2(s-1)(s+1)}=-\frac{3}{s}+\frac{2}{s-1}+\frac{2}{s+1}}$ Are you able to finish the problem? Again, if you get stuck, feel free to reply.

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