I understand that a differential equation possesses both a function and its derivative. I also understand that we solve for an unknown function.
But why do we put them in one long equation?
My basic understanding is that we do it to explain complex "things" that change over time. However, I cannot seem to understand or even phrase it better than that.
We don't do it by choice. We observe nature and notice that things are governed by differential equations. Suppose you let the water out of your bath tub. Initially the water leaves very quickly as the pressure is high. But as the water level drops, the pressure also drops and the water leaves slower. The rate of water leaving is related to the state of how much water there is. Anytime the rate of something is related to its state, you have a differential equation. This 'rate' and 'state' relation is why both a function and its derivative occur in the one equation.
The more you study differential equations, the more you will realise they are literally everywhere, and occur more than just in phenomena that 'change over time'.