Recurrence relationslinearhomogeneous Will rate immediately

Recurrence relations-linear/homogeneous

Will rate immediately

For each of the following recurrence relations determine the relation\'s order. Whether or not it is linear, and whether or not it Is homogeneous. A_n = 3A_n-1 + 2A_n-2 A_n+1 = nA_n - 4A_n-1 A_n +2 = A^2_n + 1 +2n A_k = A_k - 1 + Ak_2 + A_k-4

Solution

a)

An depends on 2 lower terms so 2nd order

Linear and homogeneous as each A_i terms has power 1 and there are no term other than A_n,A_n-1 etc.

b)

Again 2nd order following same reasoning as part a.

Again linear and homogeneous. THere is a variable coeffiicent in An but the equation is still linear and there are no terms other than A_n,A_n-1 so homogneous

c)

An depends on n-3th term so 3rd order

There is a 2 other than A_n and A_n-3 so inhomogeneous

And A_n,etc has power 1 so linear

d)

A_{n+1} has power 2 so non linear

THere is a factor of 2n so inhomogneous

And An+2 depends on immediately lower term so order 1

e)

All powers are equal to 1 so linear

No terms outside the A_i s so homogeneous

Ak depends on Ak-4 so 4th order