Small Engines (Lawn Mowers, etc.): Craftsman riding lawn mower, craftsman riding lawn mower, craftsman model

Question
I have a Craftsman model #917 273752 that will start only if I charge/jump the battery.  Once the engine is started I can mow between 5 to 10 minutes and the engine will start sputtering and than die or it will just die.  I have no idea what is wrong.  Do these mowers have alternators?  
Background:  I have replaced the battery 2 times this year and mower still dies within 5- to 10 minutes.  I have brought it to a small engine repair guy who insisted it was the air filter (not, because it did the same thing when I got it back) and the second time he looked at it assured me it was the battery (not, because it is still doing the same thing.  I really would appreciate any input you have.  I need to fix this.  Maybe my husband and I could fix it if you could point us in the right direction (I've already spent too much with my local engine repair guy who doesn't seem to know what he is doing).  Help! :(

Answer
It depends on what brand of engine is in this unit as to what the charging system consists of. It sounds to me like the charging system isn't working. I can tell you how to troubleshoot the engine if you have a 12 volt tester. I would also need the model number to find out what the charging system components are.