A different approach to the complex representation

(a)

Show, by verifying in the equation of motion and counting the free parameters, that the general solution to the equation of motion of a simple harmonic oscillator (SHO) can be written also as:

$$x(t) = x_{\rm eq} + \frac{1}{2}\left[\underline{A}e^{i\omega_o t} + \underline{A}^* e^{-i\omega_o t}\right]\;,$$ (1)

where $\underline{A}$ is the complex amplitude and $\underline{A}^*$, its complex conjugate.

(b)

Express $\underline{A}$ and $\underline{A}^*$ in terms of $x_o$, $v_o$, $x_{\rm eq}$, and $\omega_o$.