You should be able to carry out the steps listed in the document
Nonhomogeneous Linear Difference Equation with Constant Coefficients.
Although the word "nonhomogeneous" is in the title,
if you omit the "particular solution" step,
this document also applies to homogeneous LDEwccs.
Here is an ancient quiz on second-order linear difference equations (= dynamical systems): Old Test 3. It uses function notation rather than subscript notation. You should be adept at switching between these notations and manipulating them.
Reference: Goldberg, Introduction to Difference Equations, Chapter 3, Linear Difference Equations with Constant Coefficients, Sections 3.1-3.4