How to create bootstrap dropdown with hover animation?

Try this HTML/CSS code, should work

<ul class="nav navbar-nav">
        
        <li class="dropdown">
          <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Dropdown <span class="caret"></span></a>
          <ul class="dropdown-menu">
            <li><a href="#">Action</a></li>
            <li><a href="#">Another action</a></li>
            <li><a href="#">Something else here</a></li>
            <li role="separator" class="divider"></li>
            <li><a href="#">Separated link</a></li>
            <li role="separator" class="divider"></li>
            <li><a href="#">One more separated link</a></li>
          </ul>

CSS

.dropdown .dropdown-menu {
    -webkit-transition: all 0.3s;
    -moz-transition: all 0.3s;
    -ms-transition: all 0.3s;
    -o-transition: all 0.3s;
    transition: all 0.3s;

    max-height: 0;
    display: block;
    overflow: hidden;
    opacity: 0;
}

.dropdown:hover .dropdown-menu {
    max-height: 200px;
    opacity: 1;
}

Here is the working fiddle https://jsfiddle.net/d8tkq55k/ we are doing animation using CSS transition property.

Above answer works for Bootstrap 3, but if you are using Bootstrap 4, then to show dropdown of navbar on hover you can use below jQuery code

 $(document).ready(function(){
    $(".navbar .dropdown").hover(function() {
      $(this).find(".dropdown-toggle").dropdown("toggle");
    });  
})

Here is the complete code

<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css">


<script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" ></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.12.9/umd/popper.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" ></script>

<nav class="navbar navbar-expand-lg navbar-light bg-light">
  <a class="navbar-brand" href="#">Navbar</a>
  <button class="navbar-toggler" type="button" data-toggle="collapse" data-target="#navbarSupportedContent" aria-controls="navbarSupportedContent" aria-expanded="false" aria-label="Toggle navigation">
    <span class="navbar-toggler-icon"></span>
  </button>

  <div class="collapse navbar-collapse" id="navbarSupportedContent">
    <ul class="navbar-nav mr-auto">
      <li class="nav-item active">
        <a class="nav-link" href="#">Home <span class="sr-only">(current)</span></a>
      </li>
      <li class="nav-item">
        <a class="nav-link" href="#">Link</a>
      </li>
      <li class="nav-item dropdown">
        <a class="nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">
          Dropdown
        </a>
        <div class="dropdown-menu" aria-labelledby="navbarDropdown">
          <a class="dropdown-item" href="#">Action</a>
          <a class="dropdown-item" href="#">Another action</a>
          <div class="dropdown-divider"></div>
          <a class="dropdown-item" href="#">Something else here</a>
        </div>
      </li>
      <li class="nav-item">
        <a class="nav-link disabled" href="#">Disabled</a>
      </li>
    </ul>
    <form class="form-inline my-2 my-lg-0">
      <input class="form-control mr-sm-2" type="search" placeholder="Search" aria-label="Search">
      <button class="btn btn-outline-success my-2 my-sm-0" type="submit">Search</button>
    </form>
  </div>
</nav>

<script>
   $(document).ready(function(){
    $(".navbar .dropdown").hover(function() {
      $(this).find(".dropdown-toggle").dropdown("toggle");
    });
   
   
   })
</script>

Fiddle here: https://jsfiddle.net/gzrL3f8v/

Thanks