Computation for Proposition 14.13 on page 150

For this notebook to be evaluated correctly, the version of Mathematica must be able to recognize that the coefficient of   (x^(p + q - 1))   in   Expand[  x^(p+q-2)*(1+x+x^2)  ]   is   1

(This excludes Version 5.0 )

Thus, the version of Mathematica must return  True  as the answer to the question

TrueQ[
      Coefficient[Expand[x^(p + q - 2)*(1 + x + x^2)], 
          x^(p + q - 1)] == 1]

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

For (14.28), we introduce

PAA[zet_] = f'[zet]^p*P[f[zet]]

QAA[zet_] = f'[zet]^q*Q[f[zet]]

X[z_] = 
      (P[z]*D[Q[z], {z, 3}] + x1*D[P[z], z]*D[Q[z], {z, 2}] + 
          x2*D[P[z], {z, 2}]*D[Q[z], z] + x3*D[P[z], {z, 3}]*Q[z] + 
          x4*a[z]*P[z]*D[Q[z], z] + x5*a[z]*D[P[z], z]*Q[z] + 
          x6*D[a[z], z]*P[z]*Q[z])

aAA[zet_] = 
      f'[zet]^2*a[f[zet]] + f'''[zet]/(2  f'[zet]) - 
        (3/4)*(f''[zet]/f'[zet])^2

XAA[zet_] := 
      (PAA[zet]*D[QAA[zet], {zet, 3}] + 
          x1*D[PAA[zet], zet]*D[QAA[zet], {zet, 2}] + 
          x2*D[PAA[zet], {zet, 2}]*D[QAA[zet], zet] + 
          x3*D[PAA[zet], {zet, 3}]*QAA[zet] + 
          x4*aAA[zet]*PAA[zet]*D[QAA[zet], zet] + 
          x5*aAA[zet]*D[PAA[zet], zet]*QAA[zet] + 
          x6*D[aAA[zet], zet]*PAA[zet]*QAA[zet])

differ = XAA[zet] - f'[zet]^(p + q + 3)*X[f[zet]] // Expand

A1  =  
      Coefficient[ differ,  
        f'[zet]^(p + q - 1)*f''''[zet]*P[f[zet]]*
          Q[f[zet]] ]

A2  =  
      Coefficient[ differ,  
        f'[zet]^(p + q - 2)*f''[zet]*f'''[zet]*
          P[f[zet]]*Q[f[zet]] ]

A3  =  
      Coefficient[ differ,  
        f'[zet]^(p + q)*f'''[zet]*P[f[zet]]*
          Q'[f[zet]] ]

A4  =  
      Coefficient[ differ,  
        f'[zet]^(p + q)*f'''[zet]*P'[f[zet]]*
          Q[f[zet]] ]

A5  =  
      Coefficient[ differ, 
        f'[zet]^(p + q + 1)*f''[zet]*P[f[zet]]*
          Q''[f[zet]] ]

A6  =  
      Coefficient[ differ, 
        f'[zet]^(p + q + 1)*f''[zet]*P''[f[zet]]*
          Q[f[zet]] ]

A7  =  
      Coefficient[ differ,  
        f'[zet]^(p + q - 3)*f''[zet]^3*P[f[zet]]*Q[f[zet]] ]

A8  =  
      Coefficient[ differ,  
        f'[zet]^(p + q - 1)*f''[zet]^2*P[f[zet]]*
          Q'[f[zet]] ]

A9  =  
      Coefficient[differ,  
        f'[zet]^(p + q - 1)*f''[zet]^2*P'[f[zet]]*
          Q[f[zet]] ]

A10  =  
      Coefficient[ differ,  
        f'[zet]^(p + q + 1)*f''[zet]*P'[f[zet]]*
          Q'[f[zet]] ]

A11  =  
      Coefficient[ differ,  
        f'[zet]^(p + q + 1)*f''[zet]*a[f[zet]]*P[f[zet]]*
          Q[f[zet]] ]

Expand[
      differ - (
          A1*f'[zet]^(p + q - 1)*f''''[zet]*P[f[zet]]*
              Q[f[zet]] + 
            A2* f'[zet]^(p + q - 2)*f''[zet]*
              f'''[zet]*P[f[zet]]*Q[f[zet]] + 
            A3*f'[zet]^(p + q)*f'''[zet]*P[f[zet]]*
              Q'[f[zet]] + 
            A4*f'[zet]^(p + q)*f'''[zet]*P'[f[zet]]*
              Q[f[zet]] + 
            A5*f'[zet]^(p + q + 1)*f''[zet]*P[f[zet]]*
              Q''[f[zet]] + 
            A6*f'[zet]^(p + q + 1)*f''[zet]*
              P''[f[zet]]*Q[f[zet]] + 
            A7*f'[zet]^(p + q - 3)*f''[zet]^3*P[f[zet]]*
              Q[f[zet]] + 
            A8*f'[zet]^(p + q - 1)*f''[zet]^2*P[f[zet]]*
              Q'[f[zet]] + 
            A9*f'[zet]^(p + q - 1)*f''[zet]^2*P'[f[zet]]*
              Q[f[zet]] + 
            A10*f'[zet]^(p + q + 1)*f''[zet]*P'[f[zet]]*
              Q'[f[zet]] + 
            A11*f'[zet]^(p + q + 1)*f''[zet]*a[f[zet]]*
              P[f[zet]]*Q[f[zet]])]

Thus, when  differ = 0,  the coefficients  x1, x2, ...,  x6  must be solutions of the system

sol = 
      Solve[{ A1 == 0, A2 == 0, A3 == 0, A4 == 0, A5 == 0, A6 == 0, A7 == 0, 
          A8 == 0, A9 == 0, A10 == 0, A11 == 0 }, {x1, x2, x3, x4, x5, x6}]

Conversaely, when  the coefficients  x1, x2, ...,  x6  are defined by

x1  =  x1  /.  sol[[1]]  // Factor

x2  =  x2   /.  sol[[1]]  // Factor

x3  =  x3   /.  sol[[1]]  // Factor

x4  =  x4   /.  sol[[1]]   // Factor

x5  =  x5   /.  sol[[1]]  // Factor

x6  =  x6   /.  sol[[1]]  // Factor