I would like to ask the following question: Suppose I have a second order ODE and I split it into a system of two first order ODEs. I find the general solution of the second order ODE directly and I also computed the general solution of the system of two first order ODEs. I am asking to explain why both general solutions are equivalent.
I found that the eigenvalues as well as the solutions to the characteristic equation (when solving directly the linear second order ODE) are the same. Shall I make any statements for the arbitrary constants? What explanation shall I include for the question I stated above?
Thank you.