Initial Values

Hi, World!! Nice place here! My first post in this forum.

I've got a short question for a start.
If we wish to evaluate the constants for the general solution
$x(t)=C_1e^{-{\lambda_1}t}+C_2e^{-{\lambda_2}t}$
of this ODE:
$\ddot{x}+2{\gamma}\dot{x}+{{{\omega}_0}^2}x=0$
we can choose the initial conditions: $x(0)=x_0,\dot{x}(0)=v_0$
I cannot see at a glance why we can't choose an initial condition of acceleration and try to calculate the constants using this value. Why do we choose $x_0,v_0$ and not for example $x_0,a_0$ with $a_0={{\lambda_1}^2}C_1+{{\lambda_2}^2}C_2$?

Last edited:

It says we cannot specify acceleration with which the motion started because it is determined by the spring, once we specify $x_0$. But isn't the velocity also dependent on the properties of the spring then?