$Version

Mathematica Notebook for Chapter 16

The key formulas for the coefficients of the transformed equations are given on page 161 by Mathematica as follows.

c[m_,0][z_]:=1

cS[m_,i_][z_]:=Sum[Binomial[m-j,i-j]*(D[rho[z],{z,i-j}]/rho[z])*c[m,j][z],{j,0,i}]

alpha[0,j_][zeta_]:=1

alpha[i_,j_][zeta_]:=(Sum[alpha[i-1,k]'[zeta]-(i-1+k)(f''[zeta]/f'[zeta])*alpha[i-1,k][zeta],{k,1,j}])/;i≥1

cSS[m_,i_][zeta_]:=Sum[alpha[i-j,m-i][zeta]*(f'[zeta])^j*c[m,j][f[zeta]],{j,0,i}]

cS[m,1][z]

cSS[m,1][zeta]

cS[m,2][z]

Example 16.1

The following are the evaluations for Example 16.1 on page 162.

P[z_]:=c[m,2][z]+r1*c[m,1][z]^2+r2*c[m,1]'[z]

PS[z_]:=cS[m,2][z]+r1*cS[m,1][z]^2+r2*cS[m,1]'[z]

dif1[z_]=Expand[PS[z]-P[z]]

b[1]=c[m,1][z]*rho'[z]/rho[z];

b[2]=(rho'[z]/rho[z])^2;

b[3]=rho''[z]/rho[z];

a[1]=Coefficient[dif1[z],b[1]];

a[2]=Coefficient[dif1[z],b[2]];

a[3]=Coefficient[dif1[z],b[3]];

list1={a[1]==0,a[2]==0,a[3]==0}

Solve[list1,{r1,r2}]

Example 16.2

The following are the evaluations for Example 16.2 on page 143.

Q[z_]:=c[m,2][z]+s1*c[m,1][z]^2+s2*c[m,1]'[z]

QSS[zeta_]:=(cSS[m,2][zeta]+s1*cSS[m,1][zeta]^2+s2*cSS[m,1]'[zeta])

dif2[zeta_]=Expand[QSS[zeta]-(f'[zeta])^2*Q[f[zeta]]]

b[4]=c[m,1][f[zeta]] f''[zeta];

b[5]=(f''[zeta]/f'[zeta])^2;

b[6]=f'''[zeta]/f'[zeta];

a[4]=Coefficient[dif2[zeta],b[4]]

a[5]=Coefficient[dif2[zeta],b[5]]

a[6]=Coefficient[dif2[zeta],b[6]]

Solve[{a[4]==0,a[5]==0,a[6]==0},{s1,s2}]

Example 16.3

Simplify[(cS[4,4][z]-(1/4)cS[4,1][z]*cS[4,3][z]-(1/2)cS[4,3]'[z]-(9/100)cS[4,2][z]^2+(1/5)cS[4,2]''[z]+(13/100)cS[4,1][z]^2*cS[4,2][z]+(27/100)cS[4,1]'[z]*cS[4,2][z]+(1/4)cS[4,1][z]*cS[4,2]'[z]-(39/1600)cS[4,1][z]^4-(39/200)cS[4,1][z]^2*cS[4,1]'[z]-(33/200)(cS[4,1]'[z])^2-(3/20)cS[4,1][z]*cS[4,1]''[z]-(1/20)cS[4,1]'''[z])-(c[4,4][z]-(1/4)c[4,1][z]*c[4,3][z]-(1/2)c[4,3]'[z]-(9/100)c[4,2][z]^2+(1/5)c[4,2]''[z]+(13/100)c[4,1][z]^2*c[4,2][z]+(27/100)c[4,1]'[z]*c[4,2][z]+(1/4)c[4,1][z]*c[4,2]'[z]-(39/1600)c[4,1][z]^4-(39/200)c[4,1][z]^2*c[4,1]'[z]-(33/200)(c[4,1]'[z])^2-(3/20)c[4,1][z]*c[4,1]''[z]-(1/20)c[4,1]'''[z])]

Simplify[(cSS[4,4][zeta]-(1/4)cSS[4,1][zeta]*cSS[4,3][zeta]-(1/2)cSS[4,3]'[zeta]-(9/100)cSS[4,2][zeta]^2+(1/5)cSS[4,2]''[zeta]+(13/100)cSS[4,1][zeta]^2*cSS[4,2][zeta]+(27/100)cSS[4,1]'[zeta]*cSS[4,2][zeta]+(1/4)cSS[4,1][zeta]*cSS[4,2]'[zeta]-(39/1600)cSS[4,1][zeta]^4-(39/200)cSS[4,1][zeta]^2*cSS[4,1]'[zeta]-(33/200)(cSS[4,1]'[zeta])^2-(3/20)cSS[4,1][zeta]*cSS[4,1]''[zeta]-(1/20)cSS[4,1]'''[zeta])-(f'[zeta])^4(c[4,4][f[zeta]]-(1/4)c[4,1][f[zeta]]*c[4,3][f[zeta]]-(1/2)c[4,3]'[f[zeta]]-(9/100)c[4,2][f[zeta]]^2+(1/5)c[4,2]''[f[zeta]]+(13/100)c[4,1][f[zeta]]^2*c[4,2][f[zeta]]+(27/100)c[4,1]'[f[zeta]]*c[4,2][f[zeta]]+(1/4)c[4,1][f[zeta]]*c[4,2]'[f[zeta]]-(39/1600)c[4,1][f[zeta]]^4-(39/200)c[4,1][f[zeta]]^2*c[4,1]'[f[zeta]]-(33/200)(c[4,1]'[f[zeta]])^2-(3/20)c[4,1][f[zeta]]*c[4,1]''[f[zeta]]-(1/20)c[4,1]'''[f[zeta]])]

Example 16.4

The following are the evaluations for Example 4.3 on pages 145-146.

Inv[z_]:=(c[m,3][z]+t1*c[m,2][z]*c[m,1][z]+t2*c[m,1][z]^3+t3*c[m,2]'[z]+t4*c[m,1][z]*c[m,1]'[z]+t5*c[m,1]''[z])

InvS[z_]:=(cS[m,3][z]+t1*cS[m,2][z]*cS[m,1][z]+t2*cS[m,1][z]^3+t3*cS[m,2]'[z]+t4*cS[m,1][z]*cS[m,1]'[z]+t5*cS[m,1]''[z])

InvSS[zeta_]:=(cSS[m,3][zeta]+t1*cSS[m,2][zeta]*cSS[m,1][zeta]+t2*cSS[m,1][zeta]^3+t3*cSS[m,2]'[zeta]+t4*cSS[m,1][zeta]*cSS[m,1]'[zeta]+t5*cSS[m,1]''[zeta])

diff1[z_]=Expand[InvS[z]-Inv[z]]

Length[diff1[z]]

b3[1] = ((((c[m,1][z])^2) rho'[z])/(rho[z]))

b3[2] = ((c[m,2][z] rho'[z])/(rho[z]))

b3[3] = ((c[m,1][z] ((rho'[z])^2))/((rho[z])^2))

b3[4] = (((rho'[z])^3)/((rho[z])^3))

b3[5] = ((rho'[z] (c[m,1])'[z])/(rho[z]))

b3[6] = ((c[m,1][z] rho''[z])/(rho[z]))

b3[7] = ((rho'[z] rho''[z])/((rho[z])^2))

b3[8] = ((Derivative[3][rho][z])/(rho[z]))

Do[a3[k]=Coefficient[diff1[z],b3[k]],{k,1,8}]

diff2[zeta_]=Expand[InvSS[zeta]-(f'[zeta])^3*Inv[f[zeta]]]

Length[diff2[zeta]]

b3[9]= ((c[m,1][f[zeta]])^2) f'[zeta] f''[zeta]

b3[10] = c[m,2][f[zeta]] f'[zeta] f''[zeta]

b3[11] = f'[zeta] (c[m,1])'[f[zeta]] f''[zeta]

b3[12] = ((c[m,1][f[zeta]] ((f''[zeta])^2))/(f'[zeta]))

b3[13] = (((f''[zeta])^3)/((f'[zeta])^3))

b3[14] = c[m,1][f[zeta]] Derivative[3][f][zeta]

b3[15] = ((f''[zeta] Derivative[3][f][zeta])/((f'[zeta])^2))

b3[16] = ((Derivative[4][f][zeta])/(f'[zeta]))

Do[a3[k]=Coefficient[diff2[zeta],b3[k]],{k,9,16}];

sol3 = Solve[Table[a3[k]==0,{k,1,16}],{t1,t2,t3,t4,t5}]  //Factor

In case you have not obtained this result, check your definitions of  b3[1] through b3[16] and, if more detail is needed, expand this cell (by clicking at the right).   Of course, this is also the output for  the evaluations
Collect[ basicInv[m,1,3][z], list[3], Factor] 
after the other evaluations for pages 37-38.

a3[1]

a3[2]

a3[3]

a3[4]

a3[5]

a3[6]

a3[7]

a3[8]

a3[9]

a3[10]

a3[11]

a3[12]

a3[13]

a3[14]

a3[5]

a3[15]

a3[16]

Simplify[  (InvS[z] /. sol3[[1]]) - (Inv[z]  /. sol3[[1]])  ]

Simplify[  (InvSS[zeta] /. sol3[[1]]) - (f'[zeta])^3*(Inv[f[zeta]]  /. sol3[[1]])  ]

Example 16.5   (details not in monograph)

To show how the technique of Example 16.4 can be extended, we include the following computation related to the evaluation of 
Collect[ basicInv[m,1,4][z], list[4], Factor] 
after the other evaluations for pages 47-48. 
It is a bit too tedious to recommend in the monograph.

Inv4[z_]:=(c[m,4][z] + u1*c[m,1][z]*c[m,3][z] +u2*c[m,3]'[z] + u3*c[m,2][z]^2 + u4*c[m,2]''[z] + u5*c[m,1][z]^2*c[m,2][z]+ u6*c[m,1]'[z]*c[m,2][z]+u7*c[m,1][z]*c[m,2]'[z]+u8*c[m,1][z]^4+ u9*c[m,1][z]^2 *c[m,1]'[z]+ u10*c[m,1]'[z]^2 + u11*c[m,1][z] *c[m,1]''[z]+u12*c[m,1]'''[z] )

Inv4S[z_]:=(cS[m,4][z] + u1*cS[m,1][z]*cS[m,3][z] +u2*cS[m,3]'[z] + u3*cS[m,2][z]^2 + u4*cS[m,2]''[z] + u5*cS[m,1][z]^2*cS[m,2][z]+ u6*cS[m,1]'[z]*cS[m,2][z]+u7*cS[m,1][z]*cS[m,2]'[z]+u8*cS[m,1][z]^4+ u9*cS[m,1][z]^2 *cS[m,1]'[z]+ u10*cS[m,1]'[z]^2 + u11*cS[m,1][z] *cS[m,1]''[z]+u12*cS[m,1]'''[z] )

Inv4SS[zeta_]:=(cSS[m,4][zeta] + u1*cSS[m,1][zeta]*cSS[m,3][zeta] +u2*cSS[m,3]'[zeta] + u3*cSS[m,2][zeta]^2 + u4*cSS[m,2]''[zeta] + u5*cSS[m,1][zeta]^2*cSS[m,2][zeta]+ u6*cSS[m,1]'[zeta]*cSS[m,2][zeta]+u7*cSS[m,1][zeta]*cSS[m,2]'[zeta]+u8*cSS[m,1][zeta]^4+ u9*cSS[m,1][zeta]^2 *cSS[m,1]'[zeta]+ u10*cSS[m,1]'[zeta]^2 + u11*cSS[m,1][zeta] *cSS[m,1]''[zeta]+u12*cSS[m,1]'''[zeta] )

differ1[z_] = Expand[ Inv4S[z] - Inv4[z] ]

b4[1] = ((((c[m,1][z])^3) rho'[z])/(rho[z]));

b4[2] = ((c[m,1][z] c[m,2][z] rho'[z])/(rho[z]));

b4[3] = ((c[m,3][z] rho'[z])/(rho[z]));

b4[4] = ((((c[m,1][z])^2) ((rho'[z])^2))/((rho[z])^2));

b4[5] = ((c[m,2][z] ((rho'[z])^2))/((rho[z])^2));

b4[6] = ((c[m,1][z] ((rho'[z])^3))/((rho[z])^3));

b4[7] = (((rho'[z])^4)/((rho[z])^4));

b4[8] = ((c[m,1][z] rho'[z] (c[m,1])'[z])/(rho[z]));

b4[9] = ((((rho'[z])^2) (c[m,1])'[z])/((rho[z])^2));

b4[10] = ((rho'[z] (c[m,2])'[z])/(rho[z]));

b4[11] = ((((c[m,1][z])^2) rho''[z])/(rho[z]));

b4[12] = ((c[m,2][z] rho''[z])/(rho[z]));

b4[13] = ((c[m,1][z] rho'[z] rho''[z])/((rho[z])^2));

b4[14] = ((((rho'[z])^2) rho''[z])/((rho[z])^3));

b4[15] = (((c[m,1])'[z] rho''[z])/(rho[z]));

b4[16] = (((rho''[z])^2)/((rho[z])^2));

b4[17] = ((rho'[z] (c[m,1])''[z])/(rho[z]));

b4[18] = ((c[m,1][z] Derivative[3][rho][z])/(rho[z]));

b4[19] = ((rho'[z] Derivative[3][rho][z])/((rho[z])^2));

b4[20] = ((Derivative[4][rho][z])/(rho[z]));

Do[a4[k]=Coefficient[differ1[z],b4[k]],{k,1,20}]

differ2[zeta_]=Expand[Inv4SS[zeta]-(f'[zeta])^4*Inv4[f[zeta]]]

b4[21] = ((c[m,1][f[zeta]])^3) ((f'[zeta])^2) f''[zeta];

b4[22] = c[m,1][f[zeta]] c[m,2][f[zeta]] ((f'[zeta])^2) f''[zeta];

b4[23] = c[m,3][f[zeta]] ((f'[zeta])^2) f''[zeta];

b4[24] =  c[m,1][f[zeta]] ((f'[zeta])^2) (c[m,1])'[f[zeta]] f''[zeta];

b4[25] = ((f'[zeta])^2) (c[m,2])'[f[zeta]] f''[zeta];

b4[26] = ((c[m,1][f[zeta]])^2) ((f''[zeta])^2);

b4[27] = c[m,2][f[zeta]] ((f''[zeta])^2);

b4[28] = (c[m,1])'[f[zeta]] ((f''[zeta])^2);

b4[29] = ((c[m,1][f[zeta]] ((f''[zeta])^3))/((f'[zeta])^2));

b4[30] = (((f''[zeta])^4)/((f'[zeta])^4));

b4[31] = ((f'[zeta])^2) f''[zeta] (c[m,1])''[f[zeta]];

b4[32] = ((c[m,1][f[zeta]])^2) f'[zeta] Derivative[3][f][zeta];

b4[33] =  c[m,2][f[zeta]] f'[zeta] Derivative[3][f][zeta];

b4[34] = f'[zeta] (c[m,1])'[f[zeta]] Derivative[3][f][zeta];

b4[35] = ((c[m,1][f[zeta]] f''[zeta] Derivative[3][f][zeta])/(f'[zeta]));

b4[36] = ((((f''[zeta])^2) Derivative[3][f][zeta])/((f'[zeta])^3));

b4[37] = (((Derivative[3][f][zeta])^2)/((f'[zeta])^2));

b4[38] = c[m,1][f[zeta]] Derivative[4][f][zeta];

b4[39] = ((f''[zeta] Derivative[4][f][zeta])/((f'[zeta])^2));

b4[40] = ((Derivative[5][f][zeta])/(f'[zeta]));

Do[a4[k]=Coefficient[differ2[zeta],b4[k]],{k,21,40}];

sol4 = Solve[Table[a4[k]==0,{k,1,40}],{u1,u2,u3,u4,u5, u6, u7, u8, u9, u10, u11, u12}]  //Factor

Simplify[  (Inv4S[z] /. sol4[[1]]) - (Inv4[z]  /. sol4[[1]])  ]

Simplify[  (Inv4SS[zeta] /. sol4[[1]]) - (f'[zeta])^4*(Inv4[f[zeta]]  /. sol4[[1]])  ]