How to change a third order equation to the first order?

Question

I am working on this problem with a third order equation: $t_{n+2}=3t_{n+1}+t_n-3t_{n-1}$

I am asking to change this to a matrix and honestly I can not quite understand How the change works

Can someone help me a bit?

Answer 1

You can write:

$$x_{n+1}=\begin{pmatrix}t_{n+1}\\ t_{n}\\ t_{n-1} \end{pmatrix}$$

and we will try to find a matrix $M$ such that $x_{n+1}=M\cdot x_{n}$. Looking to the recurrence we get:

$$\begin{pmatrix}t_{n+2}\\ t_{n+1}\\ t_{n} \end{pmatrix}=\begin{pmatrix} 3&1&-3\\ 1&0&0\\ 0&1&0 \end{pmatrix}\cdot \begin{pmatrix}t_{n+1}\\ t_{n}\\ t_{n-1} \end{pmatrix}$$

Now you need a initial condition such that $t_0,t_1,t_2$ and then you have $x_0$.

After that you can use that $x_n=M^n\cdot x_0$