Harmonic Oscillator General Solution

The harmonic oscillator equation is

$$ \ddot{x} + \omega^2 x = 0 $$

We can write the general solution as

$$ x(t) = A_1\cos(\omega t) + A_2\sin(\omega t) $$

Or equivalently, we can write the general solution as

$$ x(t) = c_1 e^{i\omega t} + c_2 e^{-i\omega t} $$

To get from the complex exponential form to the trigonometric form, let

$$ c_1 = \frac{1}{2} \left( A_1 - iA_2 \right) $$ $$ c_2 = \frac{1}{2} \left( A_1 + iA_2 \right) $$

Using Euler's identity $ e^{i\theta} = \cos \theta + i\sin \theta $, we have

$$ \frac{1}{2} \left( A_1-iA_2 \right) \left( \cos(\omega t) + i\sin(\omega t) \right) + \frac{1}{2} \left( A_1+iA_2 \right) \left( \cos(\omega t) - i\sin(\omega t) \right) $$ $$ = \frac{A_1}{2}\cos(\omega t) + \frac{iA_1}{2}\sin(\omega t) - \frac{iA_2}{2}\cos(\omega t) + \frac{A_2}{2}\sin(\omega t) $$ $$ + \frac{A_1}{2}\cos(\omega t) - \frac{iA_1}{2}\sin(\omega t) + \frac{iA_2}{2}\cos(\omega t) + \frac{A_2}{2}\sin(\omega t) $$ $$ = A_1\cos(\omega t) + A_2\sin(\omega t) $$