Obtaining the expansion in (15.47) for (Subscript[D, 2])

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

The procedure of pages 167-168

Here, it is important to start a kernel in which the symbols  
     a[m,0,2][z],   Inv[m,1,1][z],    Inv[m,1,2][z],   and   Inv[m,2,2][z]
are merely names of variables to which no previons assignments have been made.

B[1, 1][z_] = w[1, 1][z] - w[0, 1][z]^2

B[1, 2][
        z_] = w[1, 2][z] - w[0, 1][z]*w[0, 2][z]

B[2, 2][z] = w[2, 2][z] - w[0, 2][z]^2

D2[z] = B[1, 1][z]*B[2, 2][z] - B[1, 2][z]^2

By solving (1.59) for (Subscript[w, 2, 2]),   and denoting it by  x[2,2][z], we obtain

x[2, 2][
          z_] = (Inv[m, 2, 2][
              z] + (2 (m - 1)/m)*w[0, 1][z]*w[1, 2][
                z] - (6 (m - 1)/(5 (m + 1) m)) w[0, 2][
                z]*w[1, 1][z] + w[0, 2][
                z]^2 - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) (w[0, 
                      1][z])^2*w[1, 1][
                z] - (2 (5  m + 
                      2) (m - 1)/(5 (m + 1) m)) (w[0, 
                      1][z])^2*w[0, 2][
                z] + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) (w[0, 
                      1][z])^4 + ((m - 1)/3)*
              D[w[1, 2][z], z] - ((m - 1)^2/30)*
              D[w[1, 1][z], {z, 2}] - ((m - 1)^2/(3  m))*w[
                  0, 1][z]*
              D[w[1, 1][z], 
                z] - ((5  m - 
                      4) (m - 1)^2/(15 (m + 1) m))*
              D[w[0, 1][z], z]*w[1, 1][
                z] - ((m - 1)/3)*w[0, 1][z]*
              D[w[0, 2][z], z] - ((m - 1)/3)*
              D[w[0, 1][z], z]*w[0, 2][
                z] + ((m - 1)^2/15)*w[0, 1][z]*
              D[w[0, 1][z], {z, 2}] + ((m - 1)^2/
                  15)*(D[w[0, 1][z], 
                    z])^2 + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m))*(w[0, 
                      1][z])^2*D[w[0, 1][z], z] );

x[1, 2][
          z_]  =  (Inv[m, 1, 2][
              z] + ((m - 1)/m)*w[0, 1][z]*w[1, 1][
                z] + w[0, 1][z]*w[0, 2][
                z] - ((m - 1)/
                  m) (w[0, 1][z])^3 + ((m - 1)/4)*
              D[w[1, 1][z], z] - ((m - 1)/2) w[0, 1][z]*
              D[w[0, 1][z], z]);

x[1, 1][
          z_] = Inv[m, 1, 1][z] + (w[0, 1][z])^2;

x[0, 2][z_] = ( 
          Binomial[m + 1, 3]*a[m, 2][z] + ((m - 1)/2)*
              D[w[0, 1][z], 
                z] + ((m - 1)/(2  m)) (w[0, 1][
                    z])^2 );

A1  =  
        Expand[ D2[
              z]  /.  w[2, 2][z]  ->  x[2, 2][z] ];

A2  =  
        Expand[ A1  /.  w[1, 2]'[z]  ->  x[1, 2]'[
                z] ];

A3  =  
        Expand[ A2  /.  w[1, 2][z]  ->  x[1, 2][
                z] ];

A4 =  
        Expand[ A3  /.  w[1, 1]''[z]  ->  x[1, 1]''[
                z] ];

A5 =  
        Expand[ A4  /.  w[1, 1]'[z]  ->  x[1, 1]'[
                z] ];

A6 =  
        Expand[ A5  /.  w[1, 1][z]  ->  x[1, 1][
                z] ];

A7 =  
      Expand[ FullSimplify[ 
          A6  /.  w[0, 2][z]  ->  x[0, 2][z] ] ]

Coefficient[ 
      A7,  Inv[m, 2, 2][z]*Inv[m, 1, 1][z] ]

Coefficient[ A7,  Inv[m, 1, 2][z]^2]

Coefficient[ A7,  Inv[m, 1, 1][z]*D[Inv[m, 1, 2][z], z] ]  // 
      Factor

Coefficient[ A7,  Inv[m, 1, 2][z]*D[Inv[m, 1, 1][z], z] ]  // 
      Factor

Coefficient[ 
        A7,  Inv[m, 1, 1][z]*D[Inv[m, 1, 1][z], {z, 2}] ]  // 
      Factor

Coefficient[ A7,  D[Inv[m, 1, 1][z], z]^2 ]  // 
      Factor

Coefficient[ A7,  a[m, 2][z]*Inv[m, 1, 1][z]^2 ]  // 
      Factor

And, as a final check, we use (1.14), (1.49), (1.53), and (1.59) to enter

a[m, 2][
          z_] = (1/
              Binomial[m + 1, 3])*( w[0, 2][z] - ((m - 1)/2) 
                D[w[0, 1][z], 
                  z] - ((m - 1)/(2  m))*w[0, 1][
                    z]^2 );

Inv[m, 1, 1][
          z_] = w[1, 1][z] - w[0, 1][z]^2;

Inv[m, 1, 2][
          z_] = ( w[1, 2][
              z] - ((m - 1)/m)*w[0, 1][z]*w[1, 1][
                z] - w[0, 1][z]*w[0, 2][
                z] + ((m - 1)/m) w[0, 1][z]^3 - ((m - 1)/
                  4) D[w[1, 1][z], z] + ((m - 1)/2) w[0, 1][
                z]*D[w[0, 1][z], z]);

Inv[m, 2, 2][
          z_] = ( w[2, 2][
              z] - (2 (m - 1)/m) w[0, 1][z]*w[1, 2][
                z] + (6 (m - 1)/(5 (m + 1) m)) w[0, 2][
                z]*w[1, 1][z] - w[0, 2][
                z]^2 + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) w[0, 
                    1][z]^2*w[1, 1][
                z] + (2 (5  m + 
                      2) (m - 1)/(5 (m + 1) m)) w[0, 1][
                  z]^2*w[0, 2][
                z] - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) w[0, 
                    1][z]^4 - ((m - 1)/3) 
              D[ w[1, 2][z], z] + ((m - 1)^2/30) 
              D[ w[1, 1][z],  {z, 
                  2}] + ((m - 1)^2/(3  m)) w[0, 1][z]*
              D[ w[1, 1][z], 
                z] + ((5  m - 
                      4) (m - 1)^2/(15 (m + 1) m)) 
              D[w[0, 1][z], z]*w[1, 1][
                z] + ((m - 1)/3) w[0, 1][z]*
              D[w[0, 2][z], z] + ((m - 1)/3) 
              D[w[0, 1][z], z]*w[0, 2][
                z] - ((m - 1)^2/15) w[0, 1][z]*
              D[ w[0, 1][z], {z, 2}] - ((m - 1)^2/15) 
              D[ w[0, 1][z], 
                  z]^2 - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m)) w[0, 1][
                  z]^2*D[w[0, 1][z], z]);

Inv[m, 2, 2][
          z_] = ( w[2, 2][
              z] - (2 (m - 1)/m) w[0, 1][z]*w[1, 2][
                z] + (6 (m - 1)/(5 (m + 1) m)) w[0, 2][
                z]*w[1, 1][z] - w[0, 2][
                z]^2 + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) w[0, 
                    1][z]^2*w[1, 1][
                z] + (2 (5  m + 
                      2) (m - 1)/(5 (m + 1) m)) w[0, 1][
                  z]^2*w[0, 2][
                z] - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) w[0, 
                    1][z]^4 - ((m - 1)/3) 
              D[ w[1, 2][z], z] + ((m - 1)^2/30) 
              D[ w[1, 1][z],  {z, 
                  2}] + ((m - 1)^2/(3  m)) w[0, 1][z]*
              D[ w[1, 1][z], 
                z] + ((5  m - 
                      4) (m - 1)^2/(15 (m + 1) m)) 
              D[w[0, 1][z], z]*w[1, 1][
                z] + ((m - 1)/3) w[0, 1][z]*
              D[w[0, 2][z], z] + ((m - 1)/3) 
              D[w[0, 1][z], z]*w[0, 2][
                z] - ((m - 1)^2/15) w[0, 1][z]*
              D[ w[0, 1][z], {z, 2}] - ((m - 1)^2/15) 
              D[ w[0, 1][z], 
                  z]^2 - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m)) w[0, 1][
                  z]^2*D[w[0, 1][z], z]);

\(Together[
        D2[z] - \((\(Inv[m, 2, 2]\)[z]*\(Inv[m, 1, 1]\)[z] - \(Inv[m, 1, 2]\)[
                  z]^2
 + \((\((m - 1)\)/3)\)\ \(Inv[m, 1, 1]\)[z]*
                D[\(Inv[m, 1, 2]\)[z], 
                  z]
 + \((\((m - 1)\)/3)\) \((\(-3\)/2)\)\ \(Inv[m, 1, 2]\)[
                  z]*D[\(Inv[m, 1, 1]\)[z], 
                  z] + \((\((m - 1)\)^2/20)\)*\(Inv[m, 1, 1]\)[z]*
                D[\(Inv[m, 1, 1]\)[z], {z, 2}] + \((\((m - 1)\)^2/
                    20)\) \((\(-5\)/4)\)*
                D[\(Inv[m, 1, 1]\)[z], z]^2 + \((\((m - 1)\)^2/
                    20)\) \((\(-4\)*\(a[m, 2]\)[
                      z]*\(Inv[m, 1, 1]\)[z]^2)\))\)] // Factor\)

And, the preceding checks that (15.47) of page 166 is correctly written.