Can you shoot a 38 special in a 357 desert eagle?

No. For one, the shorter .38 Special rounds won't seat in the magazine. Also, unlike revolvers, the Desert Eagle has a formed chamber in which the cartridge seats into the chamber in a manner so that proper headspace is achieved. A .38 Special cartridge will not achieve this headspace, and firing it under those conditions can be very dangerous to the firer and others.