Dynamic equations and application
Dynamic equations The simplest difference equations have the form � � = � 1 � � − 1 + � 2 � � − 2 + ⋯ + � � � � − � . The solution of this equation for x in terms of t is found by using its characteristic equation � � − � 1 � � − 1 − � 2 � � − 2 − ⋯ − � � − 1 � − � � = 0 , which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations � � − 1 = � � − 1 , … , � � − � + 1 = � � − � + 1 , giving a k -dimensional system of the first order in the stacked variable vector [ � � ⋯ � � − � + 1 ] in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots � 1 , … , � � , for use in the solution equation � � = � 1 � 1 � + ⋯ + � � � � � . A similar procedure is used for solving a differential equation of the form � �...