No it wont be covered under warranty. Capacitors are needed too keep the computer stable. While I know computers can run for a while with some failed capacitors generally it will cause failure of capacitors all around the board which can lead to hardware failures, but eventually the computer will stop booting.
If you have a really hot soldering Iron you should be able to remove the old cap and replace it. http://www.rs-components.com/index.html
you may be able to find your capacitor their. I believe its most critical to match capacitance but it may be by ohms I don't really remember.http://www.capacitorlab.com/capacitor-types-electrolytic/
This site has good instructions on how to replace a capacitor.
BTW I believe there are some sites that specialize in motherboard repair, its fairly easy to do but you would have to have soldering equipment and all that jazz which could get a bit expensive.