Linear Difference Equations

What are linear difference equations?

Linear difference equations are equations that define a sequence recursively, meaning each term in the sequence is expressed as a linear combination of previous terms. They are the discrete-time analog of linear differential equations.

General Form

A linear difference equation of order $n$ has the form:

$$ a_n y_{k+n} + a_{n-1} y_{k+n-1} + \cdots + a_1 y_{k+1} + a_0 y_k = f(k) $$

• $y_k$ is the sequence (dependent variable).
• $a_0, a_1, \ldots, a_n$ are constants (or sometimes functions of $k$).
• $f(k)$ is a known function (can be zero for homogeneous equations).

Types

1. Homogeneous linear difference equation: when $f(k) = 0$.
2. Non-homogeneous: when $f(k) \neq 0$.
3. Constant-coefficient: when all $a_i$ are constants.
4. Variable-coefficient: when some $a_i$ depend on $k$.

Examples

1. First-order homogeneous:

   $$ y_{k+1} - 3y_k = 0 $$

   Solution: $y_k = C \cdot 3^k$

2. Second-order non-homogeneous:

   $$ y_{k+2} - 4y_{k+1} + 4y_k = 2^k $$

Applications

• Time series analysis (e.g., AR models)
• Population modeling
• Control theory
• Signal processing
• Numerical algorithms

Would you like help solving a specific example?