In[]:=
ClearAll[DChange];​​DChange[expr_,transformations_List,oldVars_List,newVars_List,functions_List]:=Module[{pos,functionsReplacements,variablesReplacements,arguments,heads,newVarsSolved},pos=Flatten[Outer[Position,functions,oldVars],{{1},{2},{3,4}}];​​heads=functions[[All,0]];​​arguments=List@@@functions;​​newVarsSolved=newVars/.Solve[transformations,newVars][[1]];​​functionsReplacements=Map[Function[i,heads[[i]]->(Function[#,#2]&[arguments[[i]],ReplacePart[functions[[i]],Thread[pos[[i]]->newVarsSolved]]])],Range@Length@functions];​​variablesReplacements=Solve[transformations,oldVars][[1]];​​expr/.functionsReplacements/.variablesReplacements//Simplify//Normal];​​​​DChange[expr_,functions:{(_[___]==_)..}]:=expr/.Replace[functions,(f_[vars__]==body_):>(f->Function[{vars},body]),{1}]​​​​DChange[expr_,x___]:=DChange[expr,##]&@@Replace[{x},var:Except[_List]:>{var},{1}];​​​​DChange[expr_,coordinates:Verbatim[Rule][__String],oldVars_List,newVars_List,functions_]:=Module[{mapping,transformation},mapping=Check[CoordinateTransformData[coordinates,"Mapping",oldVars],Abort[]];​​transformation=Thread[newVars==mapping];​​{DChange[expr,transformation,oldVars,newVars,functions],transformation}];
(*---------------------------------------------------------*)
In[]:=
func=D[β[ξ],{ξ,1}]-(1/Log[f])*ω*
-β[ξ]
f
+μ+ν*
β[ξ]
f
;
In[]:=
eq=FoldDChange,func,β[ξ]==
Log[u[ξ]]
Log[f]
Log[f]u[ξ]//Expand==0
Out[]=
-ω-μu[ξ]-ν
2
u[ξ]
+
′
u
[ξ]0
In[]:=
sol1=DSolve[eq,u[ξ],ξ]
Out[]=
u[ξ]
-μ+
-
2
μ
+4νω
Tan
1
2
ξ
-
2
μ
+4νω
+
-
2
μ
+4νω

1

2ν

(*---------------------------------------------------------*)
In[]:=
sol2=FullSimplify[sol1[[1]]/.

1
->ξ0,Assumptions->{ν!=0,μ^2-4νω<0}]//Expand(*8*)
Out[]=
u[ξ]-
μ
2ν
+
-
2
μ
+4νω
Tan
1
2
(ξ+ξ0)
-
2
μ
+4νω

2ν

In[]:=
sol3=FullSimplify[sol1[[1]]/.

1
->ξ0,Assumptions->{ν!=0,μ==0,νω<0}]//Expand(*12*)
Out[]=
u[ξ]
νω
Tan(ξ+ξ0)
νω

ν
