Proof2

\[ \begin{align} y &= C_1e^{\alpha x + i\beta x} + C_2 e^{\alpha x - i\beta x} \\ \\ &= e^{\alpha x}(C_1 e^{i\beta x} + C_2 e^{-i\beta x}) \\\\ &= e^{\alpha x}\Big(C_1(\cos{\beta x} + i\sin{\beta x}) + C_2(\cos{\beta x} - i\sin{\beta x})\Big) \\\\ &= e^{\alpha x}(c_1\cos{\beta x} + c_2\sin{\beta x}) \qquad \qquad(c_1 = C_1 + C_2, \quad c_2 = C_1 - C_2) \end{align} \]

\[ \begin{gather} e^{ix} = \cos x + i\sin x\\\\\ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \\\\ \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \\ \end{gather} \]