A first-order linear difference equation is a type of equation that relates the value of a sequence at a particular time step to its value at the previous time step in a linear manner.
Detailed Explanation
A first-order linear difference equation generally takes the form:
- y[n] = α₁y[n − 1] + f(n)
Where:
y[n]is the value of the sequence at time step n.y[n - 1]is the value of the sequence at the previous time step (n-1).α₁is a constant coefficient.f(n)is a function of n, and represents an external input or forcing function. If f(n) = 0, the equation is called homogeneous.
To completely define the sequence y[n], you also need an initial condition, typically y[0] = v₀. This provides a starting point for the recursion.
Homogeneous vs. Non-Homogeneous
- Homogeneous: If
f(n) = 0, the equation simplifies toy[n] = α₁y[n − 1]. These are simpler to solve. An example isy[n] = 2y[n-1]withy[0] = 1. - Non-Homogeneous: If
f(n)is not zero, the equation is non-homogeneous. An example isy[n] = 0.5y[n-1] + nwithy[0] = 0.
Example
Consider the homogeneous equation:
y[n] = 0.8y[n - 1] with y[0] = 10
We can easily compute the first few terms:
y[0] = 10y[1] = 0.8 * y[0] = 0.8 * 10 = 8y[2] = 0.8 * y[1] = 0.8 * 8 = 6.4y[3] = 0.8 * y[2] = 0.8 * 6.4 = 5.12
In this case, the solution can be expressed in closed form as: y[n] = 10 * (0.8)^n
Key Characteristics
- First-Order: The equation only depends on the immediately preceding value in the sequence (i.e.,
y[n-1]). - Linear: The terms involving
y[n]andy[n-1]are linear (no exponents or other non-linear functions). - Difference Equation: It's a difference equation because it describes the difference between successive terms in the sequence.
- Initial Condition: The initial condition,
y[0], is necessary to uniquely determine the sequence.
First-order linear difference equations are fundamental in various fields, including finance, physics, and computer science, for modeling discrete-time systems.
