So I am studying spring mass systems, and I'm confused by why the solution, which is complex, becomes real. Such that
$$x(t) = C_1\cos(\omega t)+iC_2\sin(\omega t) = C_1\cos(\omega t)+C_2\sin(\omega t)$$
Why does the imaginary term just become an arbitrary constant?
Also the general solution is also simplified to:
$$x(t)=A\cos(\omega t+\phi)$$
How do you get that result?
I tried drawing sine and cosine waves to correlate them, but I don't see it. I also looked at some trig identities, but no avail.
A typical spring-mass oscillator has the formulation
$$ \ddot x +\omega^2 x = 0 $$
The general solution for this linear DE with lumped parameters is
$$ x = C_0 e^{\lambda t} $$
after substitution we have
$$ C_0\left(\lambda^2+\omega^2\right)e^{\lambda t}=0 $$
so this cancels for all $t$ iif $\lambda = \pm i \omega$ so
$$ x = C_1 e^{i\omega t}+ C_2 e^{-i\omega t} $$
here $C_1, C_2$ are suitable complex constants such that the contour conditions are obeyed. Note that $x(t)$ should be real.
Now using the de Moivre's identity
$$ x(t) = C_1(\cos(\omega t)+i\sin(\omega t))+C_2(\cos(\omega t)-i\sin(\omega t))) $$
or
$$ x(t) = (C_1+C_2)\cos(\omega t)+i(C_1-C_2)\sin(\omega t) $$
or
$$ x(t) = C_1'\cos(\omega t)+C_2'\sin(\omega t) $$
at this point defining
$$ \cos\phi_0 = \frac{C_1}{\sqrt{C_1'^2+C_2'^2}}\\ \sin\phi_0 = \frac{C_1}{\sqrt{C_2'^2+C_2'^2}} $$
we have
$$ x(t) = \sqrt{C_1'^2+C_2'^2}\left(\cos\phi_0 \cos(\omega t)+\sin\phi_0\sin(\omega t)\right) $$
etc.