Try below code
var table = $('#example').DataTable( {
"ajax": "../data/cat-data.txt",
"columns": [
{
"class": 'details-control',
"orderable": false,
"data": null,
"defaultContent": '',
"searchable": false
^------------------^ // added this to disable search on first column
},
{ "data": "Year" },
{ "data": "Make" },
{ "data": "Model" },
{ "data": "Engine" },
{ "data": "PartNumber" }
],
"order": [[1,'asc'], [2,'asc'], [3,'asc'], [4,'asc'], [5,'asc']],
"bSort": false,
"bPaginate": true,
"bLengthChange": true,
"bInfo": false,
"bAutoWidth": true,
"iCookieDuration": 60*60*24, // 1 day
} );
Please refer to this link for more details.
If you are not able to remove search then modify your code like below
$('#example thead th').each( function () {
var title = $('#example thead th').eq( $(this).index() ).text();
if($(this).index() !=0) // check if this is not first column header
$(this).html( '<input type="text" placeholder="Search '+title+'" />' );
} );