As others have said safari on the PC is dead and should not be used because apart from the fact that is its a very old and dead browser and will support little css3 it will also be a security issue because it is not maintained. Users should remove it from their system as it has no viable use these days.
Notwithstanding the above the problem with your code is that safari 5.1 on the PC doesn’t understand the new linear gradient syntax (because it wasn’t around then) and you would need to use the prefixed version.
background-image: -webkit-linear-gradient(bottom, #FFFFFF 0%, #DDDDDD 100%);
The above is tested and working in safari PC (I have it installed an an old machine that I don’t use anymore).