$ pwd
/tmode

$ mount
c:\cygwin\home\dkilroy\tmode on /tmode type user (textmode)
c:\cygwin\bin on /usr/bin type system (binmode)
c:\cygwin\bin on /bin type system (binmode,exec)
c:\cygwin\lib on /usr/lib type system (binmode)
c:\cygwin on / type system (binmode)

$ echo $CYGWIN


$ od -a correct.txt 
0000000   T   h   i   s  sp   f   i   l   e  cr  nl   s   h   o   u   l
0000020   d  sp   h   a   v   e  sp   D   O   S  sp   l   i   n   e  sp
0000040   e   n   d   i   n   g   s  cr  nl
0000051

$ cygcheck -c sed cygwin
Cygwin Package Information
Package              Version        Status
cygwin               1.5.19-4       OK
sed                  4.1.5-1        OK

$ sed s/lin/line/g < "c:\cygwin\home\dkilroy\tmode\original.txt" > posix_out.txt 

$ sed s/lin/line/g < "c:\cygwin\home\dkilroy\tmode\original.txt" > ".\dos_out.txt"

$ od -a posix_out.txt 
0000000   T   h   i   s  sp   f   i   l   e  cr  cr  nl   s   h   o   u
0000020   l   d  sp   h   a   v   e  sp   D   O   S  sp   l   i   n   e
0000040  sp   e   n   d   i   n   g   s  cr  cr  nl
0000053

$ od -a dos_out.txt 
0000000   T   h   i   s  sp   f   i   l   e  cr  nl   s   h   o   u   l
0000020   d  sp   h   a   v   e  sp   D   O   S  sp   l   i   n   e  sp
0000040   e   n   d   i   n   g   s  cr  nl
0000051