The harmonic oscillator: $\ddot x+\omega^2x=0$

Let the Lagrangian of a system be $L(x,\dot x)=\frac12m\dot x^2-\frac12kx^2$, where $m,k$ are positive constants. If $\omega=\sqrt{k/m}$ and $C_1,C_2$ denote arbitrary constants, then the solutions of Euler–Lagrange equations are: