What we are doing when we "solve the equation of motion" is looking for a "solution" [in the sense defined above] to the differential equation defined by Eq. (7). You may have heard horror stories about the difficulty of "solving differential equations," but it's really no big deal; like long division, basically you can only use a trial-and-error method: does this function have the right derivative? No? How about this one? And so on. Obviously, you can quickly learn to recognize certain functions by their derivatives; more complicated ones are harder, and it doesn't take much to stump even a seasoned veteran. The point of all this is that "solving differential equations" is a difficult and arcane art only if you want to be able to solve any differential equation; solving the few simple ones that occur over and over in physics is no more tedious than remembering multiplication tables. Some of the other commonly-occuring examples have already been mentioned.