Are there any derivatives that both capture the change in time and space?
I don't quite get why we are defining an ODE this way: Don't you have what amounts to a derivative on both sides of the equation there?
@siliangl There are derivatives in time, and there are derivatives in space. If you really want, you can write down a directional derivative along some arbitrary spacetime direction, but this generally doesn't come up in "normal" physical simulation (perhaps for more exotic phenomena like special/general relativity...)
@zbp You can think about it this way: you can think of the left-hand side as the actual derivative of the function you're holding in your hands; you can think of the right-hand side as the derivative that you want it to have. Solving this equation means finding a function that has this derivative.