Here, we provide a direct check on the correctness of (1.60), (1.61), and (1.62) that is independent of (1.49), (1.53), and (1.59); moreover, it is also independent of  formulas (2.5) and (3.9 ) for the coefficients of the transformed equations.

The version of Mathematica used to evaluate this notebook:

$VersionNumber

We substitute

y[z] = rho[z]*v[z]

in the situation  m = 2  of (1.1) to obtain the left member of (1.3) for  m = 2  written as

QStar = 
        Expand[(1/(rho[z])^2)*(D[y[z], {z, 2}]*D[y[z], {z, 2}] + 
                2*c[0, 1][z]*D[y[z], {z, 2}]*D[y[z], {z, 1}] + 
                2*c[0, 2][z]*D[y[z], {z, 2}]*
                  D[y[z], {z, 0}] + c[1, 1][z]*D[y[z], {z, 1}]*
                  D[y[z], {z, 1}] + 
                2*c[1, 2][z]*D[y[z], {z, 1}]*
                  D[y[z], {z, 0}] + c[2, 2][z]*D[y[z], {z, 0}]*
                  D[y[z], {z, 0}])];

with corresonding coefficients given by

cStar[0, 1][z] = 
      Expand[ Coefficient[QStar, D[v[z], {z, 2}]*D[v[z], {z, 1}]]/
          2]

D[cStar[0, 1][z], z]

cStar[0, 2][z] = 
      Expand[Coefficient[QStar, D[v[z], {z, 2}]*D[v[z], {z, 0}]]/
          2]

cStar[1, 1][z] = 
      Coefficient[QStar, D[v[z], {z, 1}]*D[v[z], {z, 1}]]

cStar[1, 2][z] = 
      Expand[ Coefficient[QStar, D[v[z], {z, 1}]*D[v[z], {z, 0}]]/
          2]

cStar[2, 2][z] = 
        Coefficient[QStar, D[v[z], {z, 0}]*D[v[z], {z, 0}]];

We have

Inv[2, 1, 1][
        z_] = c[1, 1][z] - (c[0, 1][z])^2

InvStar[2, 1, 1][
        z_] := cStar[1, 1][z] - (cStar[0, 1][z])^2

\(\(\(\(Inv[2, 1, 2]\)[
        z_] := \((\ \(c[1, 2]\)[
            z] - \((1/2)\) \(c[0, 1]\)[z]*\(c[1, 1]\)[z] - \(c[0, 1]\)[
              z]*\(c[0, 2]\)[
              z] + \((1/2)\) \((\(c[0, 1]\)[z])\)^3 - \((1/4)\) 
            D[\(c[1, 1]\)[z], z] + \((1/2)\)\ \(c[0, 1]\)[z]*
            D[\ \(c[0, 1]\)[z], z])\)\)\(
\)
    \)\)

InvStar[2, 1, 2][
        z_] := cStar[1, 2][
          z] - (1/2) cStar[0, 1][z]*cStar[1, 1][
            z] - cStar[0, 1][z]*cStar[0, 2][
            z] + (1/2) (cStar[0, 1][z])^3 - (1/4) 
          D[cStar[1, 1][z], z] + (1/2) cStar[0, 1][z]*
          D[ cStar[0, 1][z], z]

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

InvStar[2, 2, 2][
        z_] := (cStar[2, 2][
            z] - cStar[0, 1][z]*cStar[1, 2][
              z] + (1/5) cStar[0, 2][z]*cStar[1, 1][
              z] - (cStar[0, 2][z])^2 + (1/
                5) (cStar[0, 1][z])^2*cStar[1, 1][
              z] + (4/5) (cStar[0, 1][z])^2*cStar[0, 2][
              z] - (1/5) (cStar[0, 1][z])^4 - (1/3) 
            D[cStar[1, 2][z], z] + (1/30) 
            D[cStar[1, 1][z],  {z, 2}] + (1/6) cStar[0, 1][z]*
            D[ cStar[1, 1][z], z] + (1/15) 
            D[cStar[0, 1][z], z]*cStar[1, 1][
              z] + (1/3) cStar[0, 1][z]*
            D[cStar[0, 2][z], z] + (1/3) 
            D[cStar[0, 1][z], z]*cStar[0, 2][
              z] - (1/15) cStar[0, 1][z]*
            D[cStar[0, 1][z], {z, 2}] - (1/
                15) (D[cStar[0, 1][z], z])^2 - (2/
                5) (cStar[0, 1][z])^2*
            D[cStar[0, 1][z], z])

The computations

Expand[InvStar[2, 1, 1][z] - Inv[2, 1, 1][z]]

Expand[InvStar[2, 1, 2][z] - Inv[2, 1, 2][z]]

Expand[InvStar[2, 2, 2][z] - Inv[2, 2, 2][z]]

show that (1.60), (1.61), and (1.62) specify semi-invariants of the first kind.

The substitutions

y0 = u[zet]

y1 = (1/D[f[zet], zet])*D[u[zet], zet]

y2 = 
      Simplify[(1/(D[f[zet], zet])^2)*(D[u[zet], {zet, 2}] - 
              D[f[zet], {zet, 2}]*y1)]

correspond to (1.4) and transform  situation  m = 2  of (1.1) into the left member of (1.5) for  m = 2  written as

QSS = 
      Expand[(D[f[zet], zet])^4*( 
            y2*y2 + 2*c[0, 1][f[zet]]*y2*y1 + 
              2*c[0, 2][f[zet]]*y2*y0 + c[1, 1][f[zet]]*y1*y1 + 
              2*c[1, 2][f[zet]]*y1*y0 + c[2, 2][f[zet]]*y0*
                y0)]

with corresponding coefficients given by

cSS[0, 0][zet] = 
      Expand[Coefficient[QSS, D[u[zet], {zet, 2}]^2]]

cSS[0, 1][zet] = 
      Expand[Coefficient[QSS, D[u[zet], {zet, 2}]*D[u[zet], {zet, 1}]]/
          2]

cSS[0, 2][zet] = 
      Expand[Coefficient[QSS, D[u[zet], {zet, 2}]*D[u[zet], {zet, 0}]]/
          2]

cSS[1, 1][zet] = 
      Coefficient[QSS, D[u[zet], {zet, 1}]*D[u[zet], {zet, 1}]]

cSS[1, 2][zet] = 
      Expand[Coefficient[QSS, D[u[zet], {zet, 1}]*D[u[zet], {zet, 0}]]/
          2]

cSS[2, 2][zet] = 
      Coefficient[QSS, D[u[zet], {zet, 0}]*D[u[zet], {zet, 0}]]

We have

Inv[2, 1, 1][
        z_] = c[1, 1][z] - (c[0, 1][z])^2

InvSS[2, 1, 1][
        zet_] := (cSS[1, 1][
            zet] - (cSS[0, 1][zet])^2)

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

InvSS[2, 1, 2][
        zet_] = (cSS[1, 2][
              zet] - (1/2) cSS[0, 1][zet]*cSS[1, 1][
                zet] - cSS[0, 1][zet]*cSS[0, 2][
                zet] + (1/2) (cSS[0, 1][zet])^3 - (1/4) 
              D[cSS[1, 1][zet], zet] + (1/2) cSS[0, 1][zet]*
              D[ cSS[0, 1][zet], zet]) // Factor

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

\(\(\(\(InvSS[2, 2, 2]\)[
        zet_]\)\(:=\)\((\(cSS[2, 2]\)[
            zet] - \(cSS[0, 1]\)[zet]*\(cSS[1, 2]\)[
              zet] + \((1/5)\) \(cSS[0, 2]\)[zet]*\(cSS[1, 1]\)[
              zet] - \((\(cSS[0, 2]\)[zet])\)^2 + \((1/
                5)\) \((\(cSS[0, 1]\)[zet])\)^2*\(cSS[1, 1]\)[
              zet] + \((4/5)\) \((\(cSS[0, 1]\)[zet])\)^2*\(cSS[0, 2]\)[
              zet] - \((1/5)\) \((\(cSS[0, 1]\)[zet])\)^4 - \((1/3)\) 
            D[\(cSS[1, 2]\)[zet], zet] + \((1/30)\) 
            D[\(cSS[1, 1]\)[zet], {zet, 2}] + \((1/6)\) \(cSS[0, 1]\)[zet]*
            D[\ \(cSS[1, 1]\)[zet], zet] + \((1/15)\) 
            D[\(cSS[0, 1]\)[zet], zet]*\(cSS[1, 1]\)[
              zet] + \((1/3)\) \(cSS[0, 1]\)[zet]*
            D[\(cSS[0, 2]\)[zet], zet] + \((1/3)\) 
            D[\(cSS[0, 1]\)[zet], zet]*\(cSS[0, 2]\)[
              zet] - \((1/15)\) \(cSS[0, 1]\)[zet]*
            D[\(cSS[0, 1]\)[zet], {zet, 2}] - \((1/
                15)\) \((D[\(cSS[0, 1]\)[zet], zet])\)^2 - \((2/
                5)\) \((\(cSS[0, 1]\)[zet])\)^2*
            D[\(cSS[0, 1]\)[zet], zet])\)\(							\)\)\)

The computations

Expand[InvSS[2, 1, 1][
          zet] - (f'[zet])^2*Inv[2, 1, 1][f[zet]]]

Expand[InvSS[2, 1, 2][
          zet] - (f'[zet])^3*Inv[2, 1, 2][f[zet]]]

Expand[InvSS[2, 2, 2][
          zet] - (f'[zet])^4*Inv[2, 2, 2][f[zet]]]

show that (1.60), (1.61), and (1.62) specify semi-invariants of the second kind of respective weights 2, 3, and 4.  Consequently, Mathematica has checked that  (1.60), (1.61), and (1.62) specify relative invariants of respective weights 2, 3, and 4 for differential equations of the form (Subscript[Q, 2])= 0.