In the lecture, the “existence and the uniqueness theorem” of differential equations was mentioned. It states that on any point on the coordinate system (X0, Y0) of the differential equation y’=f(x, y) has one and only one solution through the point, meaning for any point on the direction field there is only one slope for the integral line (http://www.sosmath.com/diffeq/first/existence/existence.html). I thought it was easy to understand, almost a common sense somehow, because it is obvious that the original function only with different constant cannot touch each other, they are parallel to teach other. I believed so until the professor gave an example that seemingly broke the theorem.
The example was the function xy’=1-y, which lead to the integral of y=1-cx (c being the constant). The function y=1-cx, and the direction field corresponding to the function here is like a flower that radiate to every direction from the point (0, 1), which in the exact contrary of the theorem. The problem is the slope of the point (0, 1) clearly exists by it is certainly not unique, and any other point on the y axis clearly does not even exist, which makes the function seem to be a violation of the theorem. At the time, the function seemed flawless to me, but it clearly violated the other seemingly unbreakable theorem. Then it was explained: all the functions should be write into the stand form where in this case is not xy’=1-y but y’= (1-y)/x, which in this case makes the function of y’ not even continuous at any point on y-axis (when x=0), the entire prove process is established on the wrong foundation.
Formality has never been the focus on my journey of mathematical study. It seemed like the only ones that are important for me is the “ideas” and the thinking process. The formality slows me down, I prefer to do things in the way that suits my own thinking process. I have always been skipping the steps that I considered as nuance and formality: who cares whether it’s xy’=1-y or y’= (1-y)/x; they are the exact same thing, the only difference is the location of the “x”. In this case, however, formality became the key to the entire problem, it is almost unnoticeable, but overlooking this particular nuance apparently led to a shocking answer. After the lecture, I suddenly realized with the increase of my math level, formality becomes more and more important, because skipping steps means not considering the steps being skipped, means doing only the calculation but not the verification, which creates possible loopholes in logical process, and the logic in math should be flawless. Such technicality can create catastrophic results in applied math like engineering. For those who just started their math curriculum: Don’t skip steps, you may not be as smart as you think.picture from (http://onwisconsin.uwalumni.com/features/thinking-inside-the-box/)