JSFiddle - React, Tailwind, and code Playground

HTML

<script src="//ajax.googleapis.com/ajax/libs/jqueryui/1.10.3/jquery-ui.min.js"></script>
<link rel="stylesheet" href="http://ajax.googleapis.com/ajax/libs/jqueryui/1.10.3/themes/redmond/jquery-ui.css">
A configured date.
<br />
<input type='text' value='' id='date_start' />
<input type='button' id='go' value='update' />

<p>This input should reflect the <b>selected</b> date.<br />
    <input type='text' value='' class='DateInput' /></p>
<p>This input should reflect the <b>selected</b> date + 3months (my proposal for jqueryUi).<br />
    <input type='text' value='' class='DateInput' id='setDate2' /></p>
<p>This input shows jqueryUi's current status - ONLY using todays date.<br />
    <input type='text' value='' class='DateInput' id='setDate1' /></p>
<p>This input is what I believe it should do. Use "today" because it was null'd first or had no input value<br />
    <input type='text' value='' class='DateInput' id='setDate3' /></p>

JavaScript

$("#date_start, .DateInput").datepicker({
    dateFormat: "dd/mm/yy",
    changeMonth: true,
    changeYear: true,
    showButtonPanel: true
});

//Set the start values
$("#date_start").datepicker("setDate", "10/08/2013");
//Prevent User change input box
$("#date_start, #DateInput").keydown(function (e) {
    e.preventDefault();
});

$('#go').click(function () {
    //var startDate = $('#date_start').val();
    var startDate = $('#date_start').datepicker("getDate");
    //Set ALL the inputs to be that of the first.
    $(".DateInput").datepicker("setDate", startDate);
    
    //Set the third input to be now +3months using jqueryUI function
    $("#setDate1").datepicker("setDate", "+3m");
    //Set what I expect it to output
    startDate.setMonth(startDate.getMonth() + 3);
    $("#setDate2").datepicker("setDate", startDate);
    //Set the last input to use null first. (my proposal for the widget improvement)
    $("#setDate3").datepicker("setDate", null);
    $("#setDate3").datepicker("setDate", "+3m");
});