Remove Datepicker Function dynamically

You can try the enable/disable methods instead of using the option method:

$("#txtSearch").datepicker("enable");
$("#txtSearch").datepicker("disable");

This disables the entire textbox. So may be you can use datepicker.destroy() instead:

$(document).ready(function() {
    $("#ddlSearchType").change(function() {
        if ($(this).val() == "Required Date" || $(this).val() == "Submitted Date") {
            $("#txtSearch").datepicker();
        }
        else {
            $("#txtSearch").datepicker("destroy");
        }
    }).change();
});

Demo here.