For every additional inch of diameter, the tire will cover a little more than three inches more of ground for each time it goes round and round all though the town. Your axle won't compensate for this. With a 15 on one side and a 17 on the other, every time you drive far enough for it to go around twice, your 17 is trying to drag its 15-inch partner a foot farther than said 15 wants to go.
If you're not inflicting damage on your diff, you're at least wearing out one or both of those tires way too fast.
Don above has good advice if it's RWD, but if it's 4WD that will only help if (a) you have to shift into four-wheel when you want it, and (2) you stay OUT of four-wheel until you can get matching tires on the front as well.