Question

Difference between homogeneous and nonhomogeneous recurrence relation

Answer 1

According to my book, a linear homogeneous recurrence of order k is expressed this way:
A0an+A1an−1+A2an−2+⋯+Akan−k=0
While a linear non-homogeneous recurrence of order k is this way:
A0an+A1an−1+A2an−2+⋯+Akan−k=f(n)
I hardly understand what that is supposed to mean. There is not much explanation. At first, I thought that linear homogeneous were equalities to 0 while linear non-homogeneous were equalities to something else.

Well, I was wrong because later the book says that the succession defined by cn=cn−1+4cn−3 is linear homogeneous of order 3.

Answer 2

Answer:

Depend upon order

Explanation:

Linear Homogeneous Recurrence Relations with Constant Coefficients: The equation is said to be linear homogeneous difference equation if and only if R (n) = 0 and it will be of order n. The equation is said to be linear non-homogeneous difference equation if R (n) ≠ 0.

Derivation-

A linear homogeneous recurrence of order k is expressed this way:

$A0an+A1an−1+A2an−2+⋯+Akan−k=0$

While a linear non-homogeneous recurrence of order k is this way:

$A0an+A1an−1+A2an−2+⋯+Akan−k=f(n)$

I hardly understand what that is supposed to mean. There is not much explanation. At first, I thought that linear homogeneous were equalities to 0 while linear non-homogeneous were equalities to something else.

Well, I was wrong because later the book says that the succession defined by cn=cn−1+4cn−3 is linear homogeneous of order 3.

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