Nonhomogeneous recurrence relations

[Joe626]

Homework Statement

Solve the recurrence relation a_n = 3a_n−1 −2a_n−2 +3, a₀ = a₁ = 1.

Homework Equations

a_n = general solution + particular solution

The Attempt at a Solution

I started with finding the general solution, which was easy. it ended up being A₁2ⁿ + A₀

now I am having trouble solving the particular solution.

since 3 is a constant, we use B as the particular solution this results in:

B = 3B - 2B + 3

which is where i am stuck. If anyone knows how to do this i would be very grateful :)

 

[LCKurtz]

Homework Statement

Solve the recurrence relation a_n = 3a_n−1 −2a_n−2 +3, a₀ = a₁ = 1.

Homework Equations

a_n = general solution + particular solution

The Attempt at a Solution

I started with finding the general solution, which was easy. it ended up being A₁2ⁿ + A₀

now I am having trouble solving the particular solution.

since 3 is a constant, we use B as the particular solution this results in:

B = 3B - 2B + 3

which is where i am stuck. If anyone knows how to do this i would be very grateful :)

But a constant is part of the homogeneous solution, so it can't work for the NH. So, similar to what you would do in differential equations, try multiplying by $n$ instead of just a constant. So try $a_n = Cn$ and see if that can work for a particular solution.