Adding copper wire to a solution of silver nitrate in water produces solid silver metal and a solution of copper nitrate. The chemical equation is written below.
Cu + 2AgNO3 --> 2Ag + Cu(NO3)2
The solution will turn blue colored due to the presence of copper 2+ ions in solution and you will also see silver metal deposit onto the surface of the copper wire. So assuming the reaction goes to completion, the copper nitrate will be dissolved in the water and the silver metal will be a solid sitting on the bottom of the container. Since the mixture is in two distinct phases (solid and solution), it is a heterogeneous mixture.