non homogeneous recurrence relations :(

So I have an assignment and i'm having trouble doing it. The question in hand.

1. a) Solve the recurrence relation:

a_{n} = 4a_{n-1} + 5a_{n-2 }+ (-1)^{n}(36n - 6) + ( (-1)^{n}(11n - 10) / 2^{n} )

where a_{0 }= 1 and a_{1} = 0.5

I got as far as to find the characteristic solution for the homogeneous part but have no clue how to find the particular solution for the next part.

a_{n} = 4_{n-1} + 5a_{n-2}

x^{2} -4x - 5

(x - 5)(x + 1)

b_{n} = C_{1}5^{n} + C_{2}(-1)^{n}

b) Write down the close form of the generating function of the sequence a_{n. }Maybe a hint to which direction I should go?

Thank you!

Re: non homogeneous recurrence relations :(

Hey ineedalottahelp.

What do your notes give you in terms of formulae and theorems for this kind of problem?

Re: non homogeneous recurrence relations :(

Quote:

Originally Posted by

**chiro** Hey ineedalottahelp.

What do your notes give you in terms of formulae and theorems for this kind of problem?

Basically they've told us to find b_{n }

Find p_{n} for the constant part

Combine p_{n} and b_{n}

Sub in initial conditions and solve for constant variables in p_{n }

Where i'm at with finding p_{n}

4( ((-A^{n-1})(B(n-1)-C) + ((-A^{n-1})(C(n-1)-D)(E2^{n-1})) ) + 5 ( ((A^{n-2})(B(n-2)-C) + ((-A^{n-2})(C(n-2)-D)(E2^{n-2})) ) + (-1)^{n}(36n - 6) + ( (-1)^{n}(11n - 10) / 2^{n} )

But I have no clue if that's the right direction.

Re: non homogeneous recurrence relations :(

I don't know what you're note say, so I can't really comment any further.