Introduction to diff eq 101:
[Are you in an integral calc course or a differential equations course?]
Either way, there are two different concepts that are connected by the fundamental theorem of calculus, namely integration (which means adding things up or averaging things, even infinitely many), and antidifferentiation (which means finding something whose derivative equals a given function).Unfortunately they are both usualy called just integration, and this is very confusing.The good part about integration, which is really finding a limit, is that even if you cannot find it precisely, which usually you cannot, you can still approximate it as closely as desired.
In d.e. courses this aspect is called numerical solurtion of d.e.'s.
The good part about antidifferentition, is tht if you are successful at finding the antiderivative, then it immediately gives you an exact value for the otherwise difficult limit you seek.There are several different approaches to the subject because it is so hard.
One is theoretical, the proof that under some simple hypotheses, usually continuity, that solutions exist; and under some slightly greater hypotheses, usually continuous differentiability, that the solutions are unique.
A second approach is to focus only on a small category of important but very special situations or problems, whose answers are well known, and can be written specifically in term of known functions, usully exponentials, or polynomials, or trig functions.
This is the content of a "cookbook" or computational d.e. course, a common first course, and is basic technical information also useful to all.
another is the numerical approach, i.e. finding approximate answers to d.e.'s.
The first and third approaches are really the same from different viewpoints. I.e. to prove an exact solution exists you find a sequence of approximate solutions which will converge after an infinite amount of time to an exact solution. Since theoretically the amount of time required is unimportant, there does exist an exact solution, you just do not know fully what it is right now.
numerically, you take this same sequence of approximations, and follow it out as far as your computer can do in th time you have available, and that is your approximate solution, and you also know the accuracy.
then there is a "qualitative" approach, in which, rather than an exact or approximate formula for a solution, you try to give a truthful but rather incomplete description of the exact solution. I.e. you say precise correct things about the solution, they just do not have enough detail to fuly nail it down. These techniques come from topology.
The idea is to pose questions about the solution whose answers would not change if the solutioin varis slightly, and then you vary it slightly until myou can see the answer.
In other words, you group all d.e.'s into big equivalence classes, then you try to determine wghich class your problem is in. Afterwards you look for a simple problem in th same class, and use it to determine certain general aspects of the solution of your opriginal problem.
Often this is called "linearizing" the problem. It is a first cousin to the idea of studying a curve locally by looking at its tangent line.
E.g. you focus on the singularities of the solution curves and ask how much the solution flow winds around the singularity.from this point of view, a diferential equatioin is like a lot of speed signs on a road with arrows pointing in the allowed direction of travel. A solution or integral, is a car driving down the road, always going th right direction, and always at the desired speed.
hows that?
