How to write a discrete dynamical system into first order system

Question

I need guidance on how to solve this here.

$$x_{n+1} + 3x_n - 4x_{n-1} = (\sqrt{2})^n cos \left(\frac{n\pi}{6}\right)$$ I am required to transform the above equation into a first order finite discrete system. Honestly, I do not even know how to start because I believe this is a discrete system and thus I didnt expect anything like order. I guess I am wrong! Could some body help me get started? Thank you very much.

Answer 1

The role played by continuous differential equations for continuous dynamical systems is the same role played by difference equations for discrete dynamical systems.

Let \begin{align*}x_{n-1} &= y_n \Rightarrow x_n = \ y_{n+1} \end{align*} then the system becomes; $$\left\{\begin{align*} y_{n+1} &= x_n \\ x_{n+1} &= (\sqrt{2})^n cos \left(n\frac{\pi}{6}\right) + 4y_n - 3x_n \end{align*} \right.$$ which is a first order system.

Answer 2

Define the vector $$z_n:=\left[\matrix{x_{n-1} \\ x_n}\right]$$ Then, $$z_{n+1}=\left[ \matrix{0 & 1\\ 4 & -3}\right]z_n+\left[ \matrix{0 \\ 1}\right](\sqrt{2})^n\cos\left(\frac{n\pi}{6}\right)$$