Discrete Mathematics

Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

10.3: The nonhomogeneous recurrence relation

Consider the recurrence relation:

an+C₁a_n−1=f(n), n≥1, (1) where C₁is a constant and C₁6=0.

Example

Solve the recurrence relation a_n−a_n−1=3n²and a₀=7.

10.3: The nonhomogeneous recurrence relation

Consider the recurrence relation:

an+C₁a_n−1=f(n), n≥1, (1) where C₁is a constant and C₁6=0.

Example

Solve the recurrence relation a_n−a_n−1=3n²and a₀=7.

By straightforward calculations,

a_n=a₀+ Xn

i=1

f(i).

→an=7+1

2(n)(n+1)(2n+1).

By straightforward calculations,

a_n=a₀+ Xn

i=1

f(i).

→an=7+1

2(n)(n+1)(2n+1).

Method of undetermined coefficients:

Example

Solve the recurrence relation an−3a_n−1=5(7ⁿ)and a₀=2.

Method of undetermined coefficients:

Example

Solve the recurrence relation an−3a_n−1=5(7ⁿ)and a₀=2.

Step1: Seek a particular solution - a^p_n=A(7ⁿ).

Step 2:Apply the relation to find

A= 35 4

Step 3: By the initial condition, find the general solution a_n=c(3ⁿ) +A(7ⁿ).

Step1: Seek a particular solution - a^p_n=A(7ⁿ).

Step 2:Apply the relation to find

A= 35 4

Step 3: By the initial condition, find the general solution a_n=c(3ⁿ) +A(7ⁿ).

Step1: Seek a particular solution - a^p_n=A(7ⁿ).

Step 2:Apply the relation to find

A= 35 4

Step 3: By the initial condition, find the general solution a_n=c(3ⁿ) +A(7ⁿ).

Consider the nonhomogeneous relation of order 2:

a_n+C₁a_n−1+C₂a_n−2=f(n), n≥2. (2)

In general, the solution is

a_n=a^p_n+a^h_n.

Here, a^p_nis a particular solution of equation (2) and a^h_nis the solution of the associated homogeneous equation.

Consider the nonhomogeneous relation of order 2:

a_n+C₁a_n−1+C₂a_n−2=f(n), n≥2. (2)

In general, the solution is

a_n=a^p_n+a^h_n.

Here, a^p_nis a particular solution of equation (2) and a^h_nis the solution of the associated homogeneous equation.

Theorem

(I) Consider the nonhomogeneous 1st-order relation:

a_n+C₁a_n−1=krⁿ,

where k is a constant, n≥1. If rⁿ is not a solution of the associated homogeneous relation

a_n+C₁a_n−1=0,

then a^p_n=Arⁿ,where A is a constant. When rⁿis a solution of the associated homogeneous relation, then a^p_n=Bnrⁿ,for B a constant.

Theorem

(II) Consider the nonhomogeneous 2nd-order relation:

an+C1an−1+C2an−2=krⁿ, where k is a constant. Here we find that

1 a^p_n =Arⁿ, when rⁿis not a solution of the associated homogeneous relation.

2 a^p_n =Bnrⁿ, when a^h_n=c₁rⁿ+c₂r₁ⁿ, r 6=r₁.

3 a^p_n =Cn²rⁿ, when a^h_n= (c1+c₂n)rⁿ.

Example

The Koch snowflake curve:

For n≥0, let a_ndenote the area of the Polygon P_nobtained from the original equilateral triangle after we apply n

transformations of removing the middle one-third of each side and attaching a new equilateral triangle.

Please find an.

Thank you.