Computation for Formulation 14.16 on page 153

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

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).  In 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

v[0]  =  1

v[1]  =  v1

v[2]  =  v2

v[3]  =  v3

v[4]  =  v4

v[5]  =  v5

v[6]  =  v6

v[7]  =  v7

v[8]  =  v8

v[9]  =  v9

v[10]  =  v10

v[11]  =  v11

v[12]  =  v12

v[13]  =  v13

v[14]  =  v14

v[15]  =  v15

v[16]  =  v16

v[17]  =  v17

v[18]  =  v18

v[19]  =  v19

v[20]  =  v20

v[21]  =  v21

v[22]  =  v22

v[23]  =  v23

v[24]  =  v24

v[25]  =  v25

v[26]  =  v26

v[27]  =  v27

v[28]  =  v28

v[29]  =  v29

so that (14.48) yields

\(\(VAA[zet_] = 
      \((Sum[v[i]*D[PAA[zet], {zet, i}]*D[QAA[zet], {zet, 6 - i}], {i, 0, 6}]
               + aAA[zet]*
              Sum[\ v[7 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 4 - i}], {i, 0, 4}] + 
            D[aAA[zet], zet]*
              Sum[\ v[12 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 3 - i}], {i, 0, 3}]
 + 
            D[aAA[zet], {zet, 2}]*
              Sum[v[16 + i]*D[PAA[zet], {zet, i}]*D[QAA[zet], {zet, 2 - i}], {
                  i, 0, 2}] + 
            \((aAA[zet])\)^2*
              Sum[\ v[19 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 2 - i}], {i, 0, 2}] + 
            D[aAA[zet], {zet, 3}]*
              Sum[\ v[22 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
            aAA[zet]*D[aAA[zet], zet]*
              Sum[\ v[24 + i]*D[PAA[zet], {zet, i}]*
                  D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
            v[26]*D[aAA[zet], {zet, 4}]*\ PAA[zet]*QAA[zet] + 
            v[27]*aAA[zet]*D[aAA[zet], {zet, 2}]*\ PAA[zet]*QAA[zet] + 
            v[28]*D[aAA[zet], zet]^2*PAA[zet]*QAA[zet] + 
            v[29]*aAA[zet]^3*\ PAA[zet]*QAA[zet])\) // Expand; \)\)

Length[VAA[zet]]

and

V[z_] = 
      (Sum[v[i]*D[P[z], {z, i}]*D[Q[z], {z, 6 - i}], {i, 0, 6}] + 
            a[z]*Sum[ 
                v[7 + i]*D[P[z], {z, i}]*D[Q[z], {z, 4 - i}], {i, 0, 4}] + 
            D[a[z], z]*
              Sum[ v[12 + i]*D[P[z], {z, i}]*D[Q[z], {z, 3 - i}], {i, 0, 3}]
                 + D[a[z], {z, 2}]*
              Sum[ v[16 + i]*D[P[z], {z, i}]*D[Q[z], {z, 2 - i}], {i, 0, 2}]
                 + (a[z])^2*
              Sum[ v[19 + i]*D[P[z], {z, i}]*D[Q[z], {z, 2 - i}], {i, 0, 2}]
                 + D[a[z], {z, 3}]*
              Sum[ v[22 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 1}]
                 + a[z]*D[a[z], z]*
              Sum[ v[24 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 1}]
                 + v[26]*D[a[z], {z, 4}]* P[z]*Q[z] + 
            v[27]*a[z]*D[a[z], {z, 2}]*P[z]*Q[z] + 
            v[28]*D[a[z], z]^2*P[z]*Q[z] + v[29]*a[z]^3* P[z]*Q[z]) // 
        Expand;

We introduce

differ  =   
      Expand[ VAA[zet]  -  f'[zet]^(p + q + 6)*V[f[zet]] ];

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

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

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

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

m[2] = a[f[zet]]^2 Q[f[zet]] ((f'[zet])^(4 + p + q)) P'[f[zet]] f''[zet]

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

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

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

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

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

m[5] = Q[f[zet]] ((f'[zet])^(4 + p + q)) P'[f[zet]] a''[f[zet]] f''[zet]

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

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

m[6] = P[f[zet]] ((f'[zet])^(4 + p + q)) Q'[f[zet]] a''[f[zet]] f''[zet]

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

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

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

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

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

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

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

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

m[9] =  P[f[zet]] a'[f[zet]] ((f'[zet])^(2 + p + q)) Q'[f[zet]] ((f''[zet])^2)

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

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

m[10] =  a[f[zet]] ((f'[zet])^(2 + p + q)) P'[f[zet]] Q'[f[zet]] ((f''[zet])^2)

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

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

m[11] = P[f[zet]] Q[f[zet]] ((f'[zet])^(2 + p + q)) a''[f[zet]] ((f''[zet])^2)

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

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

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

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

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

m[13] = a[f[zet]] Q[f[zet]] ((f'[zet])^(p + q)) P'[f[zet]] ((f''[zet])^3)

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

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

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

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

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

m[15] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) ((f''[zet])^4)

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

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

m[16] =  ((f'[zet])^(-2 + p + q)) P'[f[zet]] Q'[f[zet]] ((f''[zet])^4)

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

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

m[17] =  Q[f[zet]] ((f'[zet])^(-4 + p + q)) P'[f[zet]] ((f''[zet])^5)

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

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

m[18] = P[f[zet]] ((f'[zet])^(-4 + p + q)) Q'[f[zet]] ((f''[zet])^5)

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

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

m[19] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-6 + p + q)) ((f''[zet])^6)

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

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

m[20] = Q[f[zet]] a'[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] P''[f[zet]]

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

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

m[21] = a[f[zet]] ((f'[zet])^(4 + p + q)) Q'[f[zet]] f''[zet] P''[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)) ((f''[zet])^2) P''[f[zet]]

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

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

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

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

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

m[24] = Q[f[zet]] ((f'[zet])^(-2 + p + q)) ((f''[zet])^4) P''[f[zet]]

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

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

m[25] = P[f[zet]] a'[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Q''[f[zet]]

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

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

m[26] = a[f[zet]] ((f'[zet])^(4 + p + q)) P'[f[zet]] f''[zet] Q''[f[zet]]

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

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

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

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

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

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

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

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

m[29] = P[f[zet]] ((f'[zet])^(-2 + p + q)) ((f''[zet])^4) Q''[f[zet]]

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

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

m[30] = ((f'[zet])^(2 + p + q)) ((f''[zet])^2) P''[f[zet]] Q''[f[zet]]

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

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

m[31] = P[f[zet]] Q[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Derivative[3][a][f[zet]]

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

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

m[32] = a[f[zet]]^2 P[f[zet]] Q[f[zet]] ((f'[zet])^(3 + p + q)) Derivative[3][f][zet]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

m[38] = a[f[zet]] Q[f[zet]] ((f'[zet])^(1 + p + q)) P'[f[zet]] f''[zet] Derivative[3][f][zet]

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

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

m[39] = a[f[zet]] P[f[zet]] ((f'[zet])^(1 + p + q)) Q'[f[zet]] f''[zet] Derivative[3][f][zet]

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

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

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

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

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

m[41] = ((f'[zet])^(-1 + p + q)) P'[f[zet]] Q'[f[zet]] ((f''[zet])^2) Derivative[3][f][zet]

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

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

m[42] =  Q[f[zet]] ((f'[zet])^(-3 + p + q)) P'[f[zet]] ((f''[zet])^3) Derivative[3][f][zet]

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

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

m[43] = P[f[zet]] ((f'[zet])^(-3 + p + q)) Q'[f[zet]] ((f''[zet])^3) Derivative[3][f][zet]

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

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

m[44] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-5 + p + q)) ((f''[zet])^4) Derivative[3][f][zet]

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

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

m[45] = a[f[zet]] Q[f[zet]] ((f'[zet])^(3 + p + q)) P''[f[zet]] Derivative[3][f][zet]

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

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

m[46] =  ((f'[zet])^(1 + p + q)) Q'[f[zet]] f''[zet] P''[f[zet]] Derivative[3][f][zet]

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

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

m[47] =  Q[f[zet]] ((f'[zet])^(-1 + p + q)) ((f''[zet])^2) P''[f[zet]] Derivative[3][f][zet]

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

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

m[48] = a[f[zet]] P[f[zet]] ((f'[zet])^(3 + p + q)) Q''[f[zet]] Derivative[3][f][zet]

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

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

m[49] = ((f'[zet])^(1 + p + q)) P'[f[zet]] f''[zet] Q''[f[zet]] Derivative[3][f][zet]

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

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

m[50] = P[f[zet]] ((f'[zet])^(-1 + p + q)) ((f''[zet])^2) Q''[f[zet]] Derivative[3][f][zet]

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

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

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

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

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

m[52] =  a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(p + q)) ((Derivative[3][f][zet])^2)

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

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

m[53] = ((f'[zet])^(p + q)) P'[f[zet]] Q'[f[zet]] ((Derivative[3][f][zet])^2)

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

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

m[54] = Q[f[zet]] ((f'[zet])^(-2 + p + q)) P'[f[zet]] f''[zet] ((Derivative[3][f][zet])^2)

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

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

m[55] = P[f[zet]] ((f'[zet])^(-2 + p + q)) Q'[f[zet]] f''[zet] ((Derivative[3][f][zet])^2)

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

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

m[56] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-4 + p + q)) ((f''[zet])^2) ((Derivative[3][f][zet])^2)

c[56]  =  Coefficient[ differ,  m[56] ]

dif[56]  =  differ  -  s[56]  // Expand;

m[57] = Q[f[zet]] ((f'[zet])^(p + q)) P''[f[zet]] ((Derivative[3][f][zet])^2)

c[57]  =  Coefficient[ differ,  m[57] ]

dif[57]  =  differ  -  s[57]  // Expand;

m[58] =  P[f[zet]] ((f'[zet])^(p + q)) Q''[f[zet]] ((Derivative[3][f][zet])^2)

c[58]  =  Coefficient[ differ,  m[58] ]

dif[58]  =  differ  -  s[58]  // Expand;

m[59] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-3 + p + q)) ((Derivative[3][f][zet])^3)

c[59]  =  Coefficient[ differ,  m[59] ]

dif[59]  =  differ  -  s[59]  // Expand;

m[60] = a[f[zet]] Q[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Derivative[3][P][f[zet]]

c[60]  =  Coefficient[ differ,  m[60] ]

dif[60]  =  differ  -  s[60]  // Expand;

m[61] =  ((f'[zet])^(2 + p + q)) Q'[f[zet]] ((f''[zet])^2) Derivative[3][P][f[zet]]

c[61]  =  Coefficient[ differ,  m[61] ]

dif[61]  =  differ  -  s[61]  // Expand;

m[62] =  Q[f[zet]] ((f'[zet])^(p + q)) ((f''[zet])^3) Derivative[3][P][f[zet]]

c[62]  =  Coefficient[ differ,  m[62] ]

dif[62]  =  differ  -  s[62]  // Expand;

m[63] = ((f'[zet])^(4 + p + q)) f''[zet] Q''[f[zet]] Derivative[3][P][f[zet]]

c[63]  =  Coefficient[ differ,  m[63] ]

dif[63]  =  differ  -  s[63]  // Expand;

m[64] = ((f'[zet])^(3 + p + q)) Q'[f[zet]] Derivative[3][f][zet] Derivative[3][P][f[zet]]

c[64]  =  Coefficient[ differ,  m[64] ]

dif[64]  =  differ  -  s[64]  // Expand;

m[65] = Q[f[zet]] ((f'[zet])^(1 + p + q)) f''[zet] Derivative[3][f][zet] Derivative[3][P][f[zet]]

c[65]  =  Coefficient[ differ,  m[65] ]

dif[65]  =  differ  -  s[65]  // Expand;

m[66] = a[f[zet]] P[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Derivative[3][Q][f[zet]]

c[66]  =  Coefficient[ differ,  m[66] ]

dif[66]  =  differ  -  s[66]  // Expand;

m[67] = ((f'[zet])^(2 + p + q)) P'[f[zet]] ((f''[zet])^2) Derivative[3][Q][f[zet]]

c[67]  =  Coefficient[ differ,  m[67] ]

dif[67]  =  differ  -  s[67]  // Expand;

m[68] = P[f[zet]] ((f'[zet])^(p + q)) ((f''[zet])^3) Derivative[3][Q][f[zet]]

c[68]  =  Coefficient[ differ,  m[68] ]

dif[68]  =  differ  -  s[68]  // Expand;

m[69] = ((f'[zet])^(4 + p + q)) f''[zet] P''[f[zet]] Derivative[3][Q][f[zet]]

c[69]  =  Coefficient[ differ,  m[69] ]

dif[69]  =  differ  -  s[69]  // Expand;

m[70] = ((f'[zet])^(3 + p + q)) P'[f[zet]] Derivative[3][f][zet] Derivative[3][Q][f[zet]]

c[70]  =  Coefficient[ differ,  m[70] ]

dif[70]  =  differ  -  s[70]  // Expand;

m[71] = P[f[zet]] ((f'[zet])^(1 + p + q)) f''[zet] Derivative[3][f][zet] Derivative[3][Q][f[zet]]

c[71]  =  Coefficient[ differ,  m[71] ]

dif[71]  =  differ  -  s[71]  // Expand;

m[72] =  P[f[zet]] Q[f[zet]] a'[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[4][f][zet]

c[72]  =  Coefficient[ differ,  m[72] ]

dif[72]  =  differ  -  s[72]  // Expand;

m[73] = a[f[zet]] Q[f[zet]] ((f'[zet])^(2 + p + q)) P'[f[zet]] Derivative[4][f][zet]

c[73]  =  Coefficient[ differ,  m[73] ]

dif[73]  =  differ  -  s[73]  // Expand;

m[74] = a[f[zet]] P[f[zet]] ((f'[zet])^(2 + p + q)) Q'[f[zet]] Derivative[4][f][zet]

c[74]  =  Coefficient[ differ,  m[74] ]

dif[74]  =  differ  -  s[74]  // Expand;

m[75] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(p + q)) f''[zet] Derivative[4][f][zet]

c[75]  =  Coefficient[ differ,  m[75] ]

dif[75]  =  differ  -  s[75]  // Expand;

m[76] = ((f'[zet])^(p + q)) P'[f[zet]] Q'[f[zet]] f''[zet] Derivative[4][f][zet]

c[76]  =  Coefficient[ differ,  m[76] ]

dif[76]  =  differ  -  s[76]  // Expand;

m[77] = Q[f[zet]] ((f'[zet])^(-2 + p + q)) P'[f[zet]] ((f''[zet])^2) Derivative[4][f][zet]

c[77]  =  Coefficient[ differ,  m[77] ]

dif[77]  =  differ  -  s[77]  // Expand;

m[78] = P[f[zet]] ((f'[zet])^(-2 + p + q)) Q'[f[zet]] ((f''[zet])^2) Derivative[4][f][zet]

c[78]  =  Coefficient[ differ,  m[78] ]

dif[78]  =  differ  -  s[78]  // Expand;

m[79] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-4 + p + q)) ((f''[zet])^3) Derivative[4][f][zet]

c[79]  =  Coefficient[ differ,  m[79] ]

dif[79]  =  differ  -  s[79]  // Expand;

m[80] = ((f'[zet])^(2 + p + q)) Q'[f[zet]] P''[f[zet]] Derivative[4][f][zet]

c[80]  =  Coefficient[ differ,  m[80] ]

dif[80]  =  differ  -  s[80]  // Expand;

m[81] = Q[f[zet]] ((f'[zet])^(p + q)) f''[zet] P''[f[zet]] Derivative[4][f][zet]

c[81]  =  Coefficient[ differ,  m[81] ]

dif[81]  =  differ  -  s[81]  // Expand;

m[82] = ((f'[zet])^(2 + p + q)) P'[f[zet]] Q''[f[zet]] Derivative[4][f][zet]

c[82]  =  Coefficient[ differ,  m[82] ]

dif[82]  =  differ  -  s[82]  // Expand;

m[83] = P[f[zet]] ((f'[zet])^(p + q)) f''[zet] Q''[f[zet]] Derivative[4][f][zet]

c[83]  =  Coefficient[ differ,  m[83] ]

dif[83]  =  differ  -  s[83]  // Expand;

m[84] = Q[f[zet]] ((f'[zet])^(-1 + p + q)) P'[f[zet]] Derivative[3][f][zet] Derivative[4][f][zet]

c[84]  =  Coefficient[ differ,  m[84] ]

dif[84]  =  differ  -  s[84]  // Expand;

m[85] = P[f[zet]] ((f'[zet])^(-1 + p + q)) Q'[f[zet]] Derivative[3][f][zet] Derivative[4][f][zet]

c[85]  =  Coefficient[ differ,  m[85] ]

dif[85]  =  differ  -  s[85]  // Expand;

m[86] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-3 + p + q)) f''[zet] Derivative[3][f][zet] Derivative[4][f][zet]

c[86]  =  Coefficient[ differ,  m[86] ]

dif[86]  =  differ  -  s[86]  // Expand;

m[87] = Q[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[3][P][f[zet]] Derivative[4][f][zet]

c[87]  =  Coefficient[ differ,  m[87] ]

dif[87]  =  differ  -  s[87]  // Expand;

m[88] =  P[f[zet]] ((f'[zet])^(2 + p + q)) Derivative[3][Q][f[zet]] Derivative[4][f][zet]

c[88]  =  Coefficient[ differ,  m[88] ]

dif[88]  =  differ  -  s[88]  // Expand;

m[89] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) ((Derivative[4][f][zet])^2)

c[89]  =  Coefficient[ differ,  m[89] ]

dif[89]  =  differ  -  s[89]  // Expand;

m[90] = ((f'[zet])^(4 + p + q)) Q'[f[zet]] f''[zet] Derivative[4][P][f[zet]]

c[90]  =  Coefficient[ differ,  m[90] ]

dif[90]  =  differ  -  s[90]  // Expand;

m[91] =  Q[f[zet]] ((f'[zet])^(2 + p + q)) ((f''[zet])^2) Derivative[4][P][f[zet]]

c[91]  =  Coefficient[ differ,  m[91] ]

dif[91]  =  differ  -  s[91]  // Expand;

m[92] =  Q[f[zet]] ((f'[zet])^(3 + p + q)) Derivative[3][f][zet] Derivative[4][P][f[zet]]

c[92]  =  Coefficient[ differ,  m[92] ]

dif[92]  =  differ  -  s[92]  // Expand;

m[93] = ((f'[zet])^(4 + p + q)) P'[f[zet]] f''[zet] Derivative[4][Q][f[zet]]

c[93]  =  Coefficient[ differ,  m[93] ]

dif[93]  =  differ  -  s[93]  // Expand;

m[94] = P[f[zet]] ((f'[zet])^(2 + p + q)) ((f''[zet])^2) Derivative[4][Q][f[zet]]

c[94]  =  Coefficient[ differ,  m[94] ]

dif[94]  =  differ  -  s[94]  // Expand;

m[95] = P[f[zet]] ((f'[zet])^(3 + p + q)) Derivative[3][f][zet] Derivative[4][Q][f[zet]]

c[95]  =  Coefficient[ differ,  m[95] ]

dif[95]  =  differ  -  s[95]  // Expand;

m[96] = a[f[zet]] P[f[zet]] Q[f[zet]] ((f'[zet])^(1 + p + q)) Derivative[5][f][zet]

c[96]  =  Coefficient[ differ,  m[96] ]

dif[96]  =  differ  -  s[96]  // Expand;

m[97] = ((f'[zet])^(1 + p + q)) P'[f[zet]] Q'[f[zet]] Derivative[5][f][zet]

c[97]  =  Coefficient[ differ,  m[97] ]

dif[97]  =  differ  -  s[97]  // Expand;

m[98] =  Q[f[zet]] ((f'[zet])^(-1 + p + q)) P'[f[zet]] f''[zet] Derivative[5][f][zet]

c[98]  =  Coefficient[ differ,  m[98] ]

dif[98]  =  differ  -  s[98]  // Expand;

m[99] = P[f[zet]] ((f'[zet])^(-1 + p + q)) Q'[f[zet]] f''[zet] Derivative[5][f][zet]

c[99]  =  Coefficient[ differ,  m[99] ]

dif[99]  =  differ  -  s[99]  // Expand;

m[100] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-3 + p + q)) ((f''[zet])^2) Derivative[5][f][zet]

c[100]  =  Coefficient[ differ,  m[100] ]

dif[100]  =  differ  -  s[100]  // Expand;

m[101] = Q[f[zet]] ((f'[zet])^(1 + p + q)) P''[f[zet]] Derivative[5][f][zet]

c[101]  =  Coefficient[ differ,  m[101] ]

dif[101]  =  differ  -  s[101]  // Expand;

m[102] = P[f[zet]] ((f'[zet])^(1 + p + q)) Q''[f[zet]] Derivative[5][f][zet]

c[102]  =  Coefficient[ differ,  m[102] ]

dif[102]  =  differ  -  s[102]  // Expand;

m[103] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) Derivative[3][f][zet] Derivative[5][f][zet]

c[103]  =  Coefficient[ differ,  m[103] ]

dif[103]  =  differ  -  s[103]  // Expand;

m[104] = Q[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Derivative[5][P][f[zet]]

c[104]  =  Coefficient[ differ,  m[104] ]

dif[104]  =  differ  -  s[104]  // Expand;

m[105] =  P[f[zet]] ((f'[zet])^(4 + p + q)) f''[zet] Derivative[5][Q][f[zet]]

c[105]  =  Coefficient[ differ,  m[105] ]

dif[105]  =  differ  -  s[105]  // Expand;

m[106] = Q[f[zet]] ((f'[zet])^(p + q)) P'[f[zet]] Derivative[6][f][zet]

c[106]  =  Coefficient[ differ,  m[106] ]

dif[106]  =  differ  -  s[106]  // Expand;

m[107] = P[f[zet]] ((f'[zet])^(p + q)) Q'[f[zet]] Derivative[6][f][zet]

c[107]  =  Coefficient[ differ,  m[107] ]

dif[107]  =  differ  -  s[107]  // Expand;

m[108] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-2 + p + q)) f''[zet] Derivative[6][f][zet]

c[108]  =  Coefficient[ differ,  m[108] ]

dif[108]  =  differ  -  s[108]  // Expand;

m[109] = P[f[zet]] Q[f[zet]] ((f'[zet])^(-1 + p + q)) Derivative[7][f][zet]

c[109]  =  Coefficient[ differ,  m[109] ]

dif[109]  =  differ  -  s[109]  // Expand

Thus, we have

Expand[  differ  -  Sum[ c[i]*m[i],  {i, 1, 109}] ]

eqs  =  Table[ c[i] ==  0,  {i, 1, 109} ];

unknowns  =  Table[ v[i],  {i, 1, 29}]

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

v1  =  v1   /.  sol[[1]]

v2  =  v2  /.  sol[[1]]  // Factor

v3  =  v3  /.  sol[[1]]

v4  =  v4  /.  sol[[1]]  // Factor

v5  =  v5  /.  sol[[1]]  // Factor

v6  =  v6  /.  sol[[1]]  // Factor

v7  =  v7  /.  sol[[1]]  // Factor

v8  =  v8  /.  sol[[1]]  // Factor

v9  =  v9  /.  sol[[1]]  // Factor

v10  =  v10  /.  sol[[1]]  // Factor

v11  =  v11  /.  sol[[1]]  // Factor

v12  =  v12  /.  sol[[1]]   // Factor

v13  =  v13  /.  sol[[1]]   // Factor

v14  =  v14  /.  sol[[1]]   // Factor

v15  =  v15  /.  sol[[1]]   // Factor

v16  =  v16  /.  sol[[1]]   // Factor

v17  =  v17  /.  sol[[1]]   // Factor

v18  =  v18  /.  sol[[1]]   // Factor

v19  =  v19  /.  sol[[1]]   // Factor

v20  =  v20  /.  sol[[1]]   // Factor

v21 =  v21  /.  sol[[1]]   // Factor

v22 =  v22  /.  sol[[1]]   // Factor

v23 =  v23  /.  sol[[1]]   // Factor

v24 =  v24  /.  sol[[1]]   // Factor

v25 =  v25  /.  sol[[1]]   // Factor

v26 =  v26  /.  sol[[1]]   // Factor

v27 =  v27  /.  sol[[1]]   // Factor

v28 =  v28  /.  sol[[1]]   // Factor

v29 =  v29  /.  sol[[1]]   // Factor

Date[]

differ // FullSimplify

Date[]

and see that  differ = 0.

V[z]

This yields (14.48) of page 153.