In[]:=
$Version
Out[]=
14.1.0 for Microsoft Windows (64-bit) (July 16, 2024)
In[]:=
eq=D[β[ξ],{ξ,1}](1/Log[f])*(ω*
-β[ξ]
f
+μ+ν*
β[ξ]
f
);
In[]:=
sol1=DSolveChangeVariablesDSolve[eq,β[ξ],ξ],u,ξ,β[ξ]==
Log[u[ξ]]
Log[f]
//FullSimplify
Out[]=
DSolve
ω+u[ξ](μ+νu[ξ])-
′
u
[ξ]
Log[f]u[ξ]
0,u[ξ],ξ
sol2=FullSimplify[DSolve[sol1[[1]],u[ξ],ξ]/.

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[DSolve[sol1[[1]],u[ξ],ξ]/.

1
->ξ0,Assumptions->{ν≠0,μ==0,νω<0}]//Expand(*12*)
Out[]=
u[ξ]
νω
Tan(ξ+ξ0)
νω

ν
