Computation for Formulation 14.15 on page 152

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 evaluates this notebook:

$VersionNumber

From (14.19), we have

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

The preceding represents  \(TraditionalForm\`\(\(a\^ ** \)\_\(m, 0, 2\)\)(ζ)\)  and we have chosen  A  to suggest asterisk.

We write

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

for (14.20).  Ind particular, we have

D[PAA[zet],  zet]

for (14.21) and

Collect[ D[PAA[zet], {zet, 2}], 
       {P''[f[zet]],  P'[f[zet]], P[f[zet]]}, Simplify ]

for (14.22).  Similarly, we introduce

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

We introduce the notation

z[0]  =  1

z[1]  =  z1

z[2]  =  z2

z[3]  =  z3

z[4]  =  z4

z[5]  =  z5

z[6]  =  z6

z[7]  =  z7

z[8]  =  z8

z[9]  =  z9

z[10]  =  z10

z[11]  =  z11

z[12]  =  z12

z[13]  =  z13

z[14]  =  z14

z[15]  =  z15

z[16]  =  z16

z[17]  =  z17

z[18]  =  z18

so that (14.47) on page 152 yields

ZAA[zet_] = 
      (Sum[  
              z[i]*D[PAA[zet], {zet, i}]*D[QAA[zet], {zet, 5 - i}], {i, 0, 5}]
               + aAA[zet]*
              Sum[ z[6 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 3 - i}], {i, 0, 3}] + 
            D[aAA[zet], zet]*
              Sum[ z[10 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 2 - i}], {i, 0, 2}] + 
            D[aAA[zet], {zet, 2}]*
              Sum[ z[13 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
            (aAA[zet])^2*
              Sum[ z[15 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
            z[17] D[aAA[zet], {zet, 3}]* PAA[zet]*QAA[zet] + 
            z[18]*aAA[zet]*D[aAA[zet], zet]*PAA[zet]*QAA[zet]) // Expand;

and

Z[z_] = 
      ( Sum[z[i]*D[P[z], {z, i}]*D[Q[z], {z, 5 - i}], {i, 0, 5}] + 
          a[z]*Sum[ z[6 + i]*D[P[z], {z, i}]*D[Q[z], {z, 3 - i}], {i, 0, 3}]
               + D[a[z], z]*
            Sum[ z[10 + i]*D[P[z], {z, i}]*D[Q[z], {z, 2 - i}], {i, 0, 2}] + 
          D[a[z], {z, 2}]*
            Sum[ z[13 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 1}] + 
          (a[z])^2*
            Sum[ z[15 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 1}] + 
          z[17]*D[a[z], {z, 3}]*P[z]*Q[z] + z[18]*a[z]*D[a[z], z]*P[z]*Q[z])

We introduce

differ  =   
      Expand[ ZAA[zet]  -  f'[zet]^(p + q + 5)*Z[f[zet]] ];

s[i_]  :=  Sum[ c[j]*m[j],  {j, 1, i}]

m[1] = a[f[zet]]^2 P[f[zet]] Q[f[zet]] ((f'[zet])^(3 + p + q)) f''[zet]

c[1]  =  Coefficient[differ,  m[1]]

dif[1]  =  differ  -  s[1]  // Expand;

m[2] =  Q[f[zet]] a'[f[zet]] ((f'[zet])^(3 + p + q)) P'[f[zet]] f''[zet]

c[2]  =  Coefficient[ differ,  m[2] ]

dif[2]  =  differ  -  s[2]  // Expand;

m[3] = P[f[zet]] a'[f[zet]] ((f'[zet])^(3 + p + q)) Q'[f[zet]] f''[zet]

c[3]  =  Coefficient[ differ,  m[3] ]

dif[3]  =  differ  -  s[3]  // Expand;

m[4] = a[f[zet]] ((f'[zet])^(3 + p + q)) P'[f[zet]] Q'[f[zet]] f''[zet]

c[4]  =  Coefficient[ differ, m[4] ]

dif[4]  =  differ  -  s[4]  // Expand;

m[5] = P[f[zet]] Q[f[zet]] ((f'[zet])^(3 + p + q)) a''[f[zet]] f''[zet]

c[5]  =  Coefficient[ differ,  m[5] ]

dif[5]  =  differ  -  s[5]  // Expand;

m[6] = P[f[zet]] Q[f[zet]] a'[f[zet]] ((f'[zet])^(1 + p + q)) ((f''[zet])^2)

c[6]  =  Coefficient[ differ,  m[6] ]

dif[6]  =  differ  -  s[6]  // Expand;

m[7] = a[f[zet]] Q[f[zet]] ((f'[zet])^(1 + p + q)) P'[f[zet]] ((f''[zet])^2)

c[7]  =  Coefficient[ differ,  m[7] ]

dif[7]  =  differ  -  s[7]  // Expand;

m[8] = a[f[zet]] P[f[zet]] ((f'[zet])^(1 + p + q)) Q'[f[zet]] ((f''[zet])^2)

c[8]  =  Coefficient[ differ,  m[8] ]

dif[8]  =  differ  -  s[8] // Expand;

m[9] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(-1 + p + q)) ((f''[zet])^3)

c[9]  =  Coefficient[ differ,  m[9] ]

dif[9]  =  differ  -  s[9]  // Expand;

m[10] = ((f'[zet])^(-1 + p + q)) P'[f[zet]] Q'[f[zet]] ((f''[zet])^3)

c[10]  =  Coefficient[ differ,  m[10] ]

dif[10]  =  differ  -  s[10]  // Expand;

m[11] = Q[f[zet]] ((f'[zet])^(-3 + p + q)) P'[f[zet]] ((f''[zet])^4)

c[11]  =  Coefficient[ differ,  m[11] ]

dif[11]  =  differ  -  s[11]  // Expand;

m[12] = P[f[zet]] ((f'[zet])^(-3 + p + q)) Q'[f[zet]] ((f''[zet])^4)

c[12]  =  Coefficient[ differ,  m[12] ]

dif[12]  =  differ  - s[12]  // Expand;

m[13] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-5 + p + q)) ((f''[zet])^5)

c[13]  =  Coefficient[ differ,  m[13] ]

dif[13]  =  differ  - s[13]  // Expand;

m[14] = a[f[zet]] Q[f[zet]] ((f'[zet])^(3 + p + q)) f''[zet] P''[f[zet]]

c[14]  =  Coefficient[ differ,  m[14] ]

dif[14]  =  differ  -  s[14]  // Expand;

m[15] = ((f'[zet])^(1 + p + q)) Q'[f[zet]] ((f''[zet])^2) P''[f[zet]]

c[15]  =  Coefficient[ differ,  m[15] ]

dif[15]  =  differ  -  s[15]  // Expand;

m[16] = Q[f[zet]] ((f'[zet])^(-1 + p + q)) ((f''[zet])^3) P''[f[zet]]

c[16]  =  Coefficient[ differ,  m[16] ]

dif[16]  =  differ  -  s[16]  // Expand;

m[17] = a[f[zet]] P[f[zet]] ((f'[zet])^(3 + p + q)) f''[zet] Q''[f[zet]]

c[17]  =  Coefficient[ differ,  m[17] ]

dif[17]  =  differ  -  s[17]  // Expand;

m[18] = ((f'[zet])^(1 + p + q)) P'[f[zet]] ((f''[zet])^2) Q''[f[zet]]

c[18]  =  Coefficient[ differ,  m[18] ]

dif[18]  =  differ  -  s[18]  // Expand;

m[19] = P[f[zet]] ((f'[zet])^(-1 + p + q)) ((f''[zet])^3) Q''[f[zet]]

c[19]  =  Coefficient[ differ,  m[19] ]

dif[19]  =  differ  -  s[19]  // Expand;

m[20] = ((f'[zet])^(3 + p + q)) f''[zet] P''[f[zet]] Q''[f[zet]]

c[20]  =  Coefficient[ differ,  m[20] ]

dif[20]  =  differ  -  s[20]  // Expand;

m[21] = P[f[zet]] Q[f[zet]] a'[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[3][f][zet]

c[21]  =  Coefficient[ differ,  m[21] ]

dif[21]  =  differ  -  s[21]  // Expand;

m[22] = a[f[zet]] Q[f[zet]] ((f'[zet])^(2 + p + q)) P'[f[zet]] Derivative[3][f][zet]

c[22]  =  Coefficient[ differ,  m[22] ]

dif[22]  =  differ  -  s[22]  // Expand;

m[23] = a[f[zet]] P[f[zet]] ((f'[zet])^(2 + p + q)) Q'[f[zet]] Derivative[3][f][zet]

c[23]  =  Coefficient[ differ,  m[23] ]

dif[23]  =  differ  -  s[23]  // Expand;

m[24] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(p + q)) f''[zet] Derivative[3][f][zet]

c[24]  =  Coefficient[ differ,  m[24] ]

dif[24]  =  differ  -  s[24]  // Expand;

m[25] = ((f'[zet])^(p + q)) P'[f[zet]] Q'[f[zet]] f''[zet] Derivative[3][f][zet]

c[25]  =  Coefficient[ differ,  m[25] ]

dif[25]  =  differ  -  s[25]  // Expand;

m[26] = Q[f[zet]] ((f'[zet])^(-2 + p + q)) P'[f[zet]] ((f''[zet])^2) Derivative[3][f][zet]

c[26]  =  Coefficient[ differ,  m[26] ]

dif[26]  =  differ  -  s[26]  // Expand;

m[27] = P[f[zet]] ((f'[zet])^(-2 + p + q)) Q'[f[zet]] ((f''[zet])^2) Derivative[3][f][zet]

c[27]  =  Coefficient[ differ,  m[27] ]

dif[27]  =  differ  -  s[27]  // Expand;

m[28] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-4 + p + q)) ((f''[zet])^3) Derivative[3][f][zet]

c[28]  =  Coefficient[ differ,  m[28] ]

dif[28]  =  differ  -  s[28]  // Expand;

m[29] = ((f'[zet])^(2 + p + q)) Q'[f[zet]] P''[f[zet]] Derivative[3][f][zet]

c[29]  =  Coefficient[ differ,  m[29] ]

dif[29]  =  differ  -  s[29]  // Expand;

m[30] = Q[f[zet]] ((f'[zet])^(p + q)) f''[zet] P''[f[zet]] Derivative[3][f][zet]

c[30]  =  Coefficient[ differ,  m[30] ]

dif[30]  =  differ  -  s[30]  // Expand;

m[31] = ((f'[zet])^(2 + p + q)) P'[f[zet]] Q''[f[zet]] Derivative[3][f][zet]

c[31]  =  Coefficient[ differ,  m[31] ]

dif[31]  =  differ  -  s[31]  // Expand;

m[32] =  P[f[zet]] ((f'[zet])^(p + q)) f''[zet] Q''[f[zet]] Derivative[3][f][zet]

c[32]  =  Coefficient[ differ,  m[32] ]

dif[32]  =  differ  -  s[32]  // Expand;

m[33] =  Q[f[zet]] ((f'[zet])^(-1 + p + q)) P'[f[zet]] ((Derivative[3][f][zet])^2)

c[33]  =  Coefficient[ differ,  m[33] ]

dif[33]  =  differ  -  s[33]  // Expand;

m[34] = P[f[zet]] ((f'[zet])^(-1 + p + q)) Q'[f[zet]] ((Derivative[3][f][zet])^2)

c[34]  =  Coefficient[ differ,  m[34] ]

dif[34]  =  differ  -  s[34]  // Expand;

m[35] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-3 + p + q)) f''[zet] ((Derivative[3][f][zet])^2)

c[35]  =  Coefficient[ differ,  m[35] ]

dif[35]  =  differ  -  s[35]  // Expand;

m[36] = ((f'[zet])^(3 + p + q)) Q'[f[zet]] f''[zet] Derivative[3][P][f[zet]]

c[36]  =  Coefficient[ differ,  m[36] ]

dif[36]  =  differ  -  s[36]  // Expand;

m[37] = Q[f[zet]] ((f'[zet])^(1 + p + q)) ((f''[zet])^2) Derivative[3][P][f[zet]]

c[37]  =  Coefficient[ differ,  m[37] ]

dif[37]  =  differ  -  s[37]  // Expand;

m[38] = Q[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[3][f][zet] Derivative[3][P][f[zet]]

c[38]  =  Coefficient[ differ,  m[38] ]

dif[38]  =  differ  -  s[38]  // Expand;

m[39] = ((f'[zet])^(3 + p + q)) P'[f[zet]] f''[zet] Derivative[3][Q][f[zet]]

c[39]  =  Coefficient[ differ,  m[39] ]

dif[39]  =  differ  -  s[39]  // Expand;

m[40] = P[f[zet]] ((f'[zet])^(1 + p + q)) ((f''[zet])^2) Derivative[3][Q][f[zet]]

c[40]  =  Coefficient[ differ,  m[40] ]

dif[40]  =  differ  -  s[40]  // Expand;

m[41] = P[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[3][f][zet] Derivative[3][Q][f[zet]]

c[41]  =  Coefficient[ differ,  m[41] ]

dif[41]  =  differ  -  s[41]  // Expand;

m[42] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(1 + p + q)) Derivative[4][f][zet]

c[42]  =  Coefficient[ differ,  m[42] ]

dif[42]  =  differ  -  s[42]  // Expand;

m[43] = ((f'[zet])^(1 + p + q)) P'[f[zet]] Q'[f[zet]] Derivative[4][f][zet]

c[43]  =  Coefficient[ differ,  m[43] ]

dif[43]  =  differ  -  s[43]  // Expand;

m[44] = Q[f[zet]] ((f'[zet])^(-1 + p + q)) P'[f[zet]] f''[zet] Derivative[4][f][zet]

c[44]  =  Coefficient[ differ,  m[44] ]

dif[44]  =  differ  -  s[44]  // Expand;

m[45] = P[f[zet]] ((f'[zet])^(-1 + p + q)) Q'[f[zet]] f''[zet] Derivative[4][f][zet]

c[45]  =  Coefficient[ differ,  m[45] ]

dif[45]  =  differ  -  s[45]  // Expand;

m[46] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-3 + p + q)) ((f''[zet])^2) Derivative[4][f][zet]

c[46]  =  Coefficient[ differ,  m[46] ]

dif[46]  =  differ  -  s[46]  // Expand;

m[47] = Q[f[zet]] ((f'[zet])^(1 + p + q)) P''[f[zet]] Derivative[4][f][zet]

c[47]  =  Coefficient[ differ,  m[47] ]

dif[47]  =  differ  -  s[47]  // Expand;

m[48] = P[f[zet]] ((f'[zet])^(1 + p + q)) Q''[f[zet]] Derivative[4][f][zet]

c[48]  =  Coefficient[ differ,  m[48] ]

dif[48]  =  differ  -  s[48]  // Expand;

m[49] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) Derivative[3][f][zet] Derivative[4][f][zet]

c[49]  =  Coefficient[ differ,  m[49] ]

dif[49]  =  differ  -  s[49]  // Expand;

m[50] = Q[f[zet]] ((f'[zet])^(3 + p + q)) f''[zet] Derivative[4][P][f[zet]]

c[50]  =  Coefficient[ differ,  m[50] ]

dif[50]  =  differ  -  s[50]  // Expand;

m[51] = P[f[zet]] ((f'[zet])^(3 + p + q)) f''[zet] Derivative[4][Q][f[zet]]

c[51]  =  Coefficient[ differ,  m[51] ]

dif[51]  =  differ  -  s[51]  // Expand;

m[52] = Q[f[zet]] ((f'[zet])^(p + q)) P'[f[zet]] Derivative[5][f][zet]

c[52]  =  Coefficient[ differ,  m[52] ]

dif[52]  =  differ  -  s[52]  // Expand;

m[53] = P[f[zet]] ((f'[zet])^(p + q)) Q'[f[zet]] Derivative[5][f][zet]

c[53]  =  Coefficient[ differ,  m[53] ]

dif[53]  =  differ  -  s[53]  // Expand;

m[54] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) f''[zet] Derivative[5][f][zet]

c[54]  =  Coefficient[ differ,  m[54] ]

dif[54]  =  differ  -  s[54]  // Expand;

m[55] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-1 + p + q)) Derivative[6][f][zet]

c[55]  =  Coefficient[ differ,  m[55] ]

dif[55]  =  differ  -  s[55]  // Expand

Thus, we have

Expand[  differ  -  Sum[ c[i]*m[i],  {i, 1, 55}] ]

eqs  =  Table[ c[i] ==  0,  {i, 1, 55} ];

unknowns  =  Table[ z[i],  {i, 1, 18}]

In order to have  differ = 0, it is necessary and sufficient that   y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11   be solutions of the system

sol  = Solve[ eqs,  unknowns]

As a check, we set

z1  =  z1   /.  sol[[1]]

z2  =  z2  /.  sol[[1]]  // Factor

z3  =  z3  /.  sol[[1]]

z4  =  z4  /.  sol[[1]]  // Factor

z5  =  z5  /.  sol[[1]]  // Factor

z6  =  z6  /.  sol[[1]]  // Factor

z7  =  z7  /.  sol[[1]]  // Factor

z8  =  z8  /.  sol[[1]]  // Factor

z9  =  z9  /.  sol[[1]]  // Factor

z10  =  z10  /.  sol[[1]]  // Factor

z11  =  z11  /.  sol[[1]]  // Factor

z12  =  z12  /.  sol[[1]]   // Factor

z13  =  z13  /.  sol[[1]]   // Factor

z14  =  z14  /.  sol[[1]]   // Factor

z15  =  z15  /.  sol[[1]]   // Factor

z16  =  z16  /.  sol[[1]]   // Factor

z17  =  z17  /.  sol[[1]]   // Factor

z18  =  z18  /.  sol[[1]]   // Factor

Date[]

differ // FullSimplify

Date[]

and see that  differ = 0.

Z[z]

It is sufficient to assume that  P  and  Q  are linearly lindependent in order to have  Z ≢ 0.