Proof2

\begin{aligned} y & = C_{1} e^{α x + i β x} + C_{2} e^{α x - i β x} \\ = e^{α x} (C_{1} e^{i β x} + C_{2} e^{- i β x}) \\ = e^{α x} (C_{1} (\cos β x + i \sin β x) + C_{2} (\cos β x - i \sin β x)) \\ = e^{α x} (c_{1} \cos β x + c_{2} \sin β x) (c_{1} = C_{1} + C_{2}, c_{2} = C_{1} - C_{2}) \end{aligned}

note :

\begin{matrix} e^{i x} = \cos x + i \sin x \\ \sin (x) = \frac{e^{i x} - e^{- i x}}{2 i} \\ \cos (x) = \frac{e^{i x} + e^{- i x}}{2} \end{matrix}