Linear Homogeneous Recurrence Relations with Constant Coefficients: Solving Recurrence Equations

Learn how to solve linear homogeneous recurrence relations with constant coefficients. This guide provides a step-by-step process, including finding the characteristic equation, determining the general solution, and applying initial conditions to find the specific solution.

Linear Homogeneous Recurrence Relations with Constant Coefficients

What are Linear Recurrence Relations?

A linear recurrence relation is an equation that defines a sequence where each term is a linear combination of preceding terms. A linear recurrence relation with constant coefficients has the form:

c₀y_n+r + c₁y_n+r-1 + c₂y_n+r-2 + ... + c_ry_n = R(n)

where c₀, c₁, ..., c_r are constants, r is the order of the recurrence relation, y_n represents the n^th term in the sequence, and R(n) is a function of n. The equation is homogeneous if R(n) = 0; otherwise, it's non-homogeneous.

Solving Linear Homogeneous Recurrence Relations

To solve a linear homogeneous recurrence relation with constant coefficients (R(n) = 0), we follow these steps:

1. Find the Characteristic Equation: Substitute y_n = Aαⁿ into the recurrence relation, where A and α are constants. This leads to the characteristic equation (a polynomial equation in α).
2. Find the Roots: Solve the characteristic equation to find the roots (α₁, α₂, ...).
3. Construct the General Solution: The form of the general solution depends on the nature of the roots:

Cases for Finding the General Solution

1. Distinct Real Roots: If there are n distinct real roots, the general solution is: y_n = A₁α₁ⁿ + A₂α₂ⁿ + ... + A_nα_nⁿ (where A₁, A₂, ... are constants).
2. Repeated Real Roots: If a root α_i is repeated k times, the solution includes terms of the form A_i1α_iⁿ + A_i2nα_iⁿ + ... + A_ikn^k-1α_iⁿ.
3. Complex Roots: If there are complex roots (α ± iβ), use Euler's formula (e^ix = cos(x) + isin(x)) to express the solution in terms of trigonometric functions.
4. Repeated Complex Roots: Similar to repeated real roots but involving trigonometric functions.

Examples: Solving Linear Homogeneous Recurrence Relations

Example 1: Distinct Real Roots

(The example showing how to solve a recurrence relation with distinct real roots is given in the original text and should be included here. The characteristic equation, its roots, and the general solution should be clearly shown.)

Example 2: Repeated Real Roots

(The example showing how to solve a recurrence relation with repeated real roots is given in the original text and should be included here. The characteristic equation, its roots, and the general solution should be clearly shown.)

Example 3: Complex Roots

(The example showing how to solve a recurrence relation with complex roots is given in the original text and should be included here. The characteristic equation, its roots, and the general solution should be clearly shown.)

Example 4: Repeated Complex Roots

(The example showing how to solve a recurrence relation with repeated complex roots is given in the original text and should be included here. The characteristic equation, its roots, and the general solution should be clearly shown.)

Conclusion

Solving linear homogeneous recurrence relations with constant coefficients is a fundamental task in discrete mathematics. The approach depends on the nature of the roots of the characteristic equation.