Hints 1 Be sure to remember what to do when there are repeat

Hints: (1) Be sure to remember what to do when there are repeated characteristic roots. (2) Try an = An + B for a particular solution

Solve a0 = 1, a1 = 6 and an = 6a n-1 - 9a n-2 + n for n > = 2 Hints: (1) Be sure to remember what to do when there are repeated characteristic roots. (2) Try an = An + B for a particular solution

Solution

for linear difference equations, we search for solutions of the form rⁿ.
so if a_n = rⁿ, the equation becomes

rⁿ=6r^n-1-9r^n-2

rⁿ=rⁿ(6/r - 9/r²)

1 = (6r-9)/r²

r² =6r-9

r²-6r+9=0

(r-3)²=0

r=3,3

So 3ⁿ and 3ⁿ are each solutions. here roots are repeated.. so the general soulution to the equation is:

Y₁ =(A+B)3ⁿ

here particular solution is given as a_n=An+B

so complete solution is a_n =y = (A+B)3ⁿ+(An+B)

n=0 =>1=A+2B=1-----1

n=1=>6=4A+4B

2A+2B=3 ----- 2

2-1 =>A=2

SUB A=2 IN 1

B=-1/2

SO closed form is

a_n = 3/2 * 3ⁿ +2n -1/2

a_n = (3ⁿ⁺¹-1)/2+2n