Probabilistic Evolution in Purely Second Order One Unknown Autonomous Explicit ODEs Under Initial Conditions

摘要

In our recent studies we have revealed that a single ODE or a set of ODEs accompanied by initial condition(s) can be taken to an infinite set of homogeneous linear ODEs with a constant infinite coefficient matrix under infinite number of compatible initial conditions. The coefficient matrix was called "Evolution Matrix" since it describes the corresponding system's evolution in time. The first order ODEs or ODE systems were considered there for simplicity despite many cases in practical applications may be related to second order ODE(s). This work extends our theory to cover the second order ODE(s). We focus on purely second order ODE cases for simplicity here. Purpose is the construction of the mainlines of the extended theory for these cases.