# Simple harmonic motion differential equation and imaginary numbers

The solution to SHM Differential Equation is $y=A\cos wt + iB\sin wt$, upon applying this, where y would be angular displacement, how can an imaginary quantity pop up when applying this to the real world? for example, at time $t=\pi/4$, you get that the angular displacement is $0.5\sqrt2+i0.5\sqrt2$, how can the angular displacement be a complex number? How can you move $i$ radians around an axis?

• You might want to this is your solution makes sense given $y$ must be real... what about taking the real part on the right? – ZeroTheHero Feb 1 '17 at 12:51
• how is that justified, to just omit the imaginary part? – Think Feb 1 '17 at 12:52
• because surely that would be a different solution of $(y=acoswt +bcoswt)$ without the i in the cos term. – Think Feb 1 '17 at 12:57
• @Think Try it without the i term. – Farcher Feb 1 '17 at 13:14

These are not the only solutions. You don't have to express solution in this way, you could just say that the solution is cos or sin without the imaginary unit $i$. BUT this solution you just wrote is a valid one. Physics do not have some kind of a monopoly to solutions of differential equations. Physicaly meaningful quantities in this solution you wrote are totally ok, you just ignore the imaginary unit and that is all.
So to sum up, in physics you are using mathematics to model reality. When you write something like this, using complex numbers, its just because it is easy to do it this way. You have to have in mind that you are taking just the real part because of course, physical quantities can not be imaginary. So if you have your oscillator with the amplitude A and some frequeny $\omega$, you can express it in many ways. The same motion.