How do we define an inertial reference frame when solving a differential equation describing the motion of an object of interest?

Question

My understanding is that, to define the reference frame under use in a differential equation describing the motion of a particle, initial conditions are required; and in fact, it is the initial conditions $x(t_0)=x_0$ and $v(t_0)=v_0$ that define, uniquely, the coordinate system in use.

I understand why the condition $x(t_0)=x_0$ is necessary, but not why $v(t_0)=v_0$ also is. I'd be grateful if you would help me out with my understanding of what exactly defines uniquely the reference frame in a problem involving the motion of particles.

Answer 1

If you do not give the velocity of the particle you did not fully defined the frame. You can have an infinite number of frames in which the particle initial position is $x_0$, but in those inertial frames that move relative to each other will see the particle as having a different initial velocity. You want to work in a single frame, so you need to chose the initial speed.

Alternatively, the reason you need to give the speed is that dynamic equations are usually of second order, and to find a particular solution of a second order differential equation you usually have to give both the initial value of the variable and the initial value of its derivative