$\sqrt{-1}$ coefficient in a function In a simple harmonic oscillator with $\ddot{x} = -\omega^2x$, it can be shown through differentiation that one solution can be given by $\dot{x} = i \omega Ae^{i \omega t}$. What does the factor of $i$  do here? What effect does it have on velocity?
 A: Note that $i$ can be written as follows
\begin{align}
i = e^{i \pi/2}.
\end{align}
Therefore, we have the velocity $\dot{x}$ written as
\begin{align}
\dot{x} \; &= \; i \omega A e^{i \omega t} \\
\; &= \; \omega A \; e^{i \pi/2} \, e^{i \omega t} \\
\; &= \; \omega A \; e^{i \big(\omega t + \pi/2\big)}.
\end{align}
If you compare the velocity $\dot{x}$ with displacement $x = A e^{i \omega t}$,  $x$ is lag behind $\dot{x}$ by a phase of $\pi/2$.
So the complex number $i$ you mentioned encodes the phase difference information.
A: $$
\ddot x + \omega^2 x =0
$$
Is a second order linear differential equation so there will be two solutions as the basis functions adding which you can get any solution. What you have to note is that the equation allows complex solutions. That doesn't mean you have to take the complex solution. In particular if you start with purely real initial conditions $x(0), v(0) \in \mathbb R$, the evolution will not make the solution develop an imaginary part. You should try to convince yourself of this using either analytical or numerical solutions.
Thus the solution you write can never be a physical solution. So your question is meaningless. You have to construct solutions that are real using the basis $e^{\pm i\omega t}$. Or you could simply work in the basis $\sin(\omega t), \cos(\omega t)$ as long as the coefficients multiplying them are real.
So the bottomline is that the mathematical equation allows non-physical solutions and it is upto the physicist to filter them to reflect experimental observations.
A: The terms in a differential equation representing an oscillating system may be out of phase. Such equations can be represented by a “phase diagram”  Each term in the equation can be considered as representing one component of a vector rotating around the origin in a 2D “phase space”. The phase space can be chosen as an xy system, but there some mathematical advantages to putting it in a complex number plane (if you are comfortable with such a system) where R$e^{iθ}$ = Rcos(θ) + iRsin(θ). (The θ's may be modified with a phase shift). Generally, the real component of each vector represents the instantaneous value of the corresponding term in the equation, and the angle represents a phase. In your equations, the multiplying, i, indicates that there is a 90 degree phase difference between the vector representing velocity and the one representing position. (The sine term becomes real and the cosine imaginary.)
