I’m having some trouble predicting the behavior of ODE’s using the mass-spring analogy. For example, consider the second order IVP listed below: $$y’’ – \space 6y’ + 8y = 0, \space \space \space \space y(0) = 2, \space \space y’(0) = -8$$ Now I believe that the initial condition of $y(0) = 2$ indicates that the original displacement of the spring from equilibrium is 2 units, but I could be wrong on that. I have no idea what $y’(0) = -8$ means intuitively, so that’s the first thing I’m hoping to get cleared up (ie interpreting the initial conditions).
Next, I understand that each coefficient means different things. From the equation above I see that there is an inertial mass of 1, a damping factor of -6, and a stiffness factor of +8. Since the damping factor is negative I think that the spring will not converge, but outside of that I’m not sure how to interpret the other coefficients. So as another question, how do you know when the solution will oscillate with increasing distance, or go off to positive/negative infinity, or simply oscillate, etc.
I'm really seeking advice on interpreting what the behavior of the spring will be for all 8 different scenarios when the coefficients are positive and negative. I’m already planning to plug in some examples using wolfram alpha to see what’s going on, but getting some deeper understanding from the forum will definitely help. Any tips and help with this would be greatly appreciated.
Edit: More specifically, I'm interested in how the solution behaves graphically over time.