You can filter the thumbnail HTML before it is printed out. If it is empty – return the placeholder.

This should go into your theme’s functions.php:

add_filter( 'post_thumbnail_html', 'wpse_63591_default_thumb' );

function wpse_63591_default_thumb( $html )
{
    if ( '' !== $html )
    {
        return '&lt;a href="' . get_permalink() . '"&gt;' . $html . '&lt;/a&gt;';
    }

    return '&lt;img src="' . get_bloginfo( 'stylesheet_directory' ) . '/images/shared/placeholder.jpg" /&gt;';    
}


In your template just use 

the_post_thumbnail('size-2');


No conditionals or complicated PHP needed anymore. :)

Setting a default placeholder image WITHOUT link
I am an new user of Mathematica and learning basics so please guide me

I tried to use this code 

Animate[Plot[x^2/a^2 - y^2/b^2 == 1, {x, 0, 2}, {y, 0, 2}], 
  {a, 1, 3}, 
  {b, 1, 3}]


but I got this error message:


  Plot::nonopt: Options expected (instead of {y,0,2}) beyond position 3 in Plot[x^2/FEa$$116^2-y^2/FEb$$116^2==1,{x,0,2},{y,0,2}]. An option must be a rule or a list of rules. >>


All I want to do is to animate a hyperbole with its asymptotes.
