Solve Difference Equations: Finding Impulse Response

[EugP]

Homework Statement

For the discrete-time system:

$$y[k+2]+\frac{1}{6}y[k+1]-\frac{1}{6}y[k]=f[k+1]+f[k]$$

Find the impulse response.

Homework Equations

The Attempt at a Solution

Alright so I started like this:

$$h_0[k+2]+\frac{1}{6}h_0[k+1]-\frac{1}{6}h_0[k]=0$$

$$h_0[1]=0$$

$$h_0[2]=1$$

Now this is where I'm stuck. I know I need to get the equation for $$h_0[k]$$, but I don't know how. The equation they got is:

$$h_0[k]=C_1(-\frac{1}{2})^k+C_2(\frac{1}{3})^k$$

Can anyone tell me how they got there?

 

[wildman]

It is the homogeneous solution of the difference equation:

You can take the characteristic equation which is a quadratic:$$m^2 + \frac{1}{6}m-\frac{1}{6}=0$$

and then take the roots. You will find the roots to be -1/2 and 1/3. The
$$C_1$$ and $$C_1$$ are constants made necessary because the ambiguity in the solution (same as differential equations). The answer is then just the roots taken to the power of k. k is just the value in a sequence.

 

[EugP]

Thank you so much, this clears everything up for me!

 

[wildman]

you are welcome