# General Process to Solve a Differential Equation

Lately in my Physics C class we have been doing differential equations, which my teacher has explained more than once to me. Yet for some reason I can still not grasp the concept, I think my problem lies in not understanding the process of solving a differential equation. So my question then becomes what is the general process for solving a differential equation of the first order? I am a high school senior and a physics major by heart, so I really need to understand this process before I go to college. I am in Calculus AB and any answers are appreciated.

• In a differential equation, the unknown is a function, and the equation involves derivatives of the unknown. As to "the general process for solving a differential equation", there's entire courses devoted to methods. Even "first order" is not really enough, as there are several kinds of "first order" equations. Linear? Separable? – Arturo Magidin Dec 14 '11 at 5:06
• Unfortunately, there is no single process to solve an arbitrary first order equation. The task of finding such a method is fairly hopeless. However, at the calculus AB level, separation of variables and substitution is usually all you need. You may want to google those terms. – user12014 Dec 14 '11 at 5:10
• And to add to the two methods mentioned, about the only other method that might come into play is that of an integrating factor. – Arturo Magidin Dec 14 '11 at 5:33

The most important things about (systems of) differential equations are the following: (i) They enable you to describe in precise mathematical terms your intuitive physical ideas about how a (physical, biological, oeconomical, $\ldots$) "system" you are studying is evolving in the next second. (ii) There is a mathematically proven "existence and uniqueness theorem" that guarantees a unique solution for all times $t\geq 0$ when you have thought long enough about your "system" – even if you are not able to write down this solution in an explicit formula.