Yep, using a pae-kernel should be possible with a pae-capable CPU. But more factors might play a roll in this. I know my old eeepc with the atom cpu has pae, but the motherboard is not suitable to use it so I'm forced to stick with a non-pae kernel. But you can try it, if it does not work you just uninstall the 3.13 kernel and headers to revert back to the 3.2 kernel you have now.
Install the patched kernel using terminology:
sudo apt-get install linux-image-extra-3.13.0-100-generic linux-headers-3.13.0-100-generic
Now Ubuntu has a boot-option for this, called forcepae that sets the missing flag.
For this you should add -- forcepae after the words quite splash in the grub kernel line. You can edit that line hitting the "e" when the grub screen appears. On a single-OS machine you might need to hold SHIFT at boot.
Now you can test the new kernel. Once you know it works you can make that option permanent by changing grub defaults.
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