Find the particular solution of this difference equation (solve most of it)

[jwxie]

Homework Statement

Find the particular solution.

y(k+2) + y(k+1) -6y(k) = 3^(k)

Homework Equations

The Attempt at a Solution

Still need to be answered

Question (1): Before finding the particular solution, is it true that we should ALWAYS get the homogeneous solution (leaving in the form of undetermined constants), to see whether the functional form of the homogeneous and particular solutions are the same.

For example:
$y(k) - 3y(k-1) - 4y(k-2) = 4^{k} + (2)(4^{k-1})$
y_h includes a solution of the form $C(4)^{k}$. Thus, we have the multiply the particular solution by k (which is the way my professor did it).


[STRIKE]Now, the real question:

(2)
I tried to solve the problem at the beginning of the thread but I couldn't get the right answer.

I know that the particular solution should be in the form of B3^(k), and according to the homogeneous solution, we have a form C(-3)^k but since it has the - sign it's a different form.

I will show my steps below:
sub the Y_particular into the original equation
$B3^{k+2}+B3^{k+1}-6B3^{k}=3^{k}$

divide by 3^{k+2}
$B+\frac{B}{3}-\frac{6}{9}=\frac{1}{9}$

Combine terms I have
$\frac{4B}{3}-\frac{6}{9}=\frac{1}{9}$
and B is 7/12

However, my teacher gave Y_p = 1/6 * 3^k as the answer...

Can anyone help me? Sorry for all the questions... thanks!

-- his answer is correct if i do the substitution...[/STRIKE]

 

[jwxie]

ha i caught my silly mistake
urgh
thanks if anyone had actually read this...

but i still hope someone can answer question 1 for me...
thanks!