Consider the horizontal spring-mass system where the spring-force is the only force acting on the mass. Suppose that a mass is initially at $x=x_0$ with an initial velocity $v_0$. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at $x=x_0$ and the other corresponding to the mass initially at the equilibrium position with velocity $v_0$. What is the amplitude of the total oscillation?
I know that $x(t)=c_1\cos \omega t+c_2\sin \omega t$ and $\frac{dx}{dt}=-c_1\omega\sin \omega t+c_2\omega\cos \omega t$. I know that at $t=0$, $x(0)=x_0=c_1$ and $v_0=c_2\omega$. I also know that when $v=0$, $c_2=0$. But now I'm completely stuck on what to do from here. Any help would be greatly appreciated!