The separation of variables is a technique for finding general solutions to a differential equation. In this context, “general solutions” means that we will be solving for a set or family of functions. This may seem like abstract wording, but this idea has already been explored in integration and antidifferentiation: 💯
This statement reads as “C is any real number, which implies f(x) + C is confined to a certain set of functions S.” Or, more simply put, this means that there are infinite solutions to Eq. 22, but the solution must include the condition that C is a constant that does not depend on x.
This means that f(x) + C is a general solution. A specific solution for this same example would be: C is a specific number. Differential equations function in a similar way. First, find a general solution, then find a specific solution if you are given more information. 📖