+Holger K. von Jouanne-Diedrich
That's a great question, which can be somewhat answered using the final couple of slides.
The derivative of a complex wave is the same wave rotated by 90 degrees—and scaled by a constant, though in my case it's 1. So if you can decompose your solution into a sum of complex waves of different positive and negative frequencies (i.e. fourier transform), then differential relationships are simpler to express.
The only catch is that different frequencies have to undergo a different phase shift in order to move the same physical distance. e.g. Two opposite frequencies have to twist in opposite directions to add up to a single real wave that moves at constant speed. And frequency 2 will move half as far if you twist it 180 degrees than frequency 1. If you don't do this, they go out of phase and add up to a random wave, as in the final slide.
Hence, if you want to fourier-transform a derivative over units of distance (uniform for all frequencies) rather than over units of phase (frequency-specific), you have to scale each complex wave's phase proportional to the frequency. But scaling phase = multiplying angles = the power of a complex number.
Hence, each application of the derivative on a complex wave multiplies by a complex constant which is power-proportional to frequency (i.e. first derivative x iɷ, second derivative x iɷ², etc.).
This is why Fourier transforms turn differential equations into polynomials. The differential equation's conditions have been transformed into a power-relation of complex numbers. You're expressing the phase conditions required for all your complex waves to add up to a coherent, real solution that satisfies the constraints.
The kicker is then that Laplace transforms (decomposing into exponentials) and Fourier transforms (decomposing into complex waves) turn out to be identical, aside from a 90 degree rotation. Because (e^x)∠y = e^(x+iy) = (cos y + i*sin y) * e^x.
Thanks to the magic of complex exponentials combining both exponential changes in magnitude and proportional changes in phase/angle, both wave-like and exponential-like differential equations can be expressed as two faces of the same coin.
Indeed, when you Laplace transform a real differential equation and try to solve the polynomial involved, you either get real roots, or pairs of complementary complex ones. Such a pair represents the fact that you always need two counter-rotating complex waves in the solution to add up to a real value.
I'm a bit rusty on this stuff though, I'm working from memory here.