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[]:=
solution=DSolve[eq,β,ξ]
DSolve
:Inverse functions are being used by DSolve, so some solutions may not be found.
Out[]=
βFunction{ξ},
Log
-μ+
-
2
μ
+4νω
Tan
1
2
ξ
-
2
μ
+4νω
+
-
2
μ
+4νω

1
Log[f]
2ν

Log[f]

eq/.solution//FullSimplify(*Ok*)
Out[]=
{True}