Continuing with the topic of the complex numbers, the professor talks about two kinds of solutions for a specific type of differential equation, y’+ky=kcosωt.
As usual, the professor solves it by imagine the left side as the real part of a function containing complex number. The solution before simplification is (1/1+(w/k))+e^iωt, and the professor expand the function by going polar, which is a complicated process involving “translating” the function into polar form, with vectors and angles. It was assumed as the standard form him. What surprised me was the second way he taught the student to use, it is called Cartesian method (More about Cartesian method please click Here, which is just to use a series of formulas and to multiply the complex number by itself, (for example use a-bi to multiply a+bi), in order to gain an rational denominator and then to purge the irrational numbers by canceling the number with numerators formed by complex numbers. This method is unbelievable easy, but the result it gives looks totally different from the answer given by the polar method. One shows a completely abstract part other than the common part ((1/(1+ω2/k2)^1/2)*cos(ωt-φ), and the Cartesian method gives an answer containing a relatively geometric part ((1/1+ω2/k2)*(cosωt+ (ω/k)sinωt)). The two answers need to agree with each other, they have to. Then, the professor gives one single basic formula: a conθ+b sinθ=C cos(θ-φ). Without going into the details, the formula connects perfectly the abstract side to the understandable geometric side. (More on the subject please click here)
The fact of using one basic, single equation to solve an extremely complicated problem has always fascinated me, it even reminded me of a movie. The Theory of Everything is a movie about how Stephen Hawking redefine the limit, both personally and academically. There is one special conversation between him and his mentor after his dissertation about the existence of a singular equation and a black hole that can be used to explain and trace the time itself to its beginning. At the time, he is already terribly influenced by the ALS, carrying two canes, and cannot even speak clearly. The mentor asked, “So what’s next?” he smiled, “Prove it. Prove, with a singular equation that time had a beginning. Wouldn’t that be nice, Professor? That simple, elegant equation that explains everything.” The world is a complicated place, just like the world of math. Sometimes, however all we need is a singular, simple equation that can connect fields like the one I learned, or achieve revolution, like so many others.