Luckly, sometimes the exact solution can be very accuately aproximated with a wave ecuation.
Luckly, sometimes the exact solution can be very accuately aproximated with a particle ecuation.
(Sometimes, the exact solution can be aproximated saying that the lowest energy state is an eigenvector of the Schoedinger equation. Is that a wave? It's not localized, but not very wavy.)
But neither are the exact solution, just aproximations that solve tpgether 99% of the experiment.
It's difficult to explain, because to explaing the detials you need like two years of algebra and calculus and then like another 2 years of physics, and now you get a degree in physics.
It's possible to solve the difficult ecuation only in very simple cases like electron-electron colissions, if you allow some cheating and a tiny error. For more complicated systems like electron-muon there are some problems. And for more complicated systems, you get more technical problems and more aproximations.