text: # Laplace transforms to solve initial value DE / partial fractions
  
  1. Sep 20, 2013
  
  ### CrazyCamo
  
  Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated.
  
  1. The problem statement, all variables and given/known data
  
  Use Laplace transforms to solve the initial value problem:
  
  f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1
  
  Solve this problem using laplace transforms, showing all steps in your reasoning. State the solution y(t) for each of 0<t<1 and t>1, then sketch it over the range 0<= t <= 10, noting its main features.
  
  2. Relevant equations
  
  3. The attempt at a solution
  
  I have gotten up to F(Y) = (e^-s + s^2 + 3s)/(s(s^2 + 4s+8))
  
  However, from here i am not sure what to do. I tried taking the partial fractions of:
  
  1/(s(s^2 + 4s+8))
  
  but am getting very confused. Again any help would be much appreciated. Cheers
  
  2. Sep 21, 2013
  
  ### SteamKing
  
  Staff Emeritus
  Why don't you break F(Y) down into the sum of its various components?
  
  E.g., D = (s(s^2+4s+8))
  
  F(Y) = (e^-s)/D + s^2/D + 3s/D
  
  You can tackle each term individually.
  
  PS: finding the PFE of 1/D doesn't help.
  
  3. Sep 21, 2013
  
  ### HallsofIvy
  
  Staff Emeritus
  The denominator is $$s(s^2+4s+ 8)= s(s^2+ 4s+ 4+ 4)= s((s+ 2)^2+ 4)$$ so you can use "partial fractions to write that as $$\frac{A}s+ \frac{Bs+ C}{(s+2)^2+ 4}$$.
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