content: Step 0 (to simplify the calculation): add short names
  
  Laplace: f'' t - c * f' t + 4 * f t = 0, f 0 = 2, f' 0 = c
    where  c = 3*r
           r = sqrt 2
  
  > module Solution_202508_Laplace where
  > import PS  -- Power Series - could use DSLsofMath.PSDS
  
  3a: Solve using power series
  
  > c :: Fractional a => a
  > c = 4.242640687119286 -- 3 * sqrt 2
  > fs, fs', fs'' :: Fractional a => PS a
  > fs'' = c*fs' - 4*fs
  > fs' = integ c fs''
  > fs  = integ 2 fs'
  
  Hand-computing these series gives us:
  
  Step 1: definition of |integ| (and ignoring the PS constructor):
  
  c = 3*r
  r = sqrt 2
  
  < fs   ~=   2 : _
  < fs'  ~=   c : _
  
  Compute the first element of |fs''|:
  
  < head fs''  =  c*c-4*2 = 10
  
  Now we can compute the 2nd elements of |fs| and |fs'| from the first
  elements of |fs'| and |fs''|:
  
  < fs    ~=   2 :  c : _
  < fs'   ~=   c : 10 : __
  
  Finally we can compute the 3rd element of |fs| from the 2nd of |fs'|:
  
  < fs    ~=   2 :  c : 5 : _
  
  And that is all the question asked for:
  
  Answer: take 3 fs ~= [2,c,5].
  
  > main = takePS 3 fs ~= PS [2,c,5]
  
  ----------------
  3b: Solve using the Laplace transform
  
  Derivative rule + initial conditions:
  
  L f' s = -f 0 + s * L f s = -2 + s * L f s
  L f'' s = -f' 0 - s*f 0 + s^2*L f s = -c - 2*s + s^2*L f s
  
  The Laplace transform of the equation thus gives
  
    (-c - 2*s + s^2*L f s)-c*(-2 + s * L f s)+4*L f s = 0
  
  Collecting terms:
  
    (-c - 2*s +2*c) + (s^2-c*s+4)*L f s  = 0
  
  Factoring and moving terms to make the L-term stand alone.
  
    (s-r)*(s-2*r)*L f s  = 2*s - 3*r
  
  Now use the Ansatz  L f s = A/(s-r) + B/(s-2*r)
  
    A*(s-2*r) + B*(s-r) = 2*s - 3*r
  
  Solve for A and B (for example by inspection).
  
    A=1, B=1
  
  Thus
  
    L f s = 1/(s-r) + 1/(s-2r)
  
  Reverse transform by linearity and the rule in the question.
  
    f t = exp(r*t) + exp(2*r*t)
  
  Check the initial conditions:
  
    f 0 = 1 + 1 = 2 -- OK
  
    f' t = r*exp(r*t) + 2*r*exp(2*r*t)
    f' 0 = r + 2*r = 3*r = c -- OK
  
  Check the full equation:
  
    f'' t = r^2*exp(r*t) + 4*r^2*exp(2*r*t)
          =   2*exp(r*t) +     8*exp(2*r*t)
  
    f'' t - c * f' t + 4 * f t
  = -- expand the definitions + use r^2 = 2
      2*exp(r*t) + 8*exp(2*r*t)
    - 3*r*(r*exp(r*t) + 2*r*exp(2*r*t))
    + 4*(exp(r*t) + exp(2*r*t))
  = -- collect the coefficients
    (2-6+4)*exp(r*t) + (8-12+4)*exp(2*r*t)
  = -- simple arithmetics
    0
  
  OK!
  