Substitution for impulse in recurrence equation

[yoran]

Hi,

I was wondering what substitution to try when finding a particular solution for a recurrence equation with a linear combination of impulses on the right side of the equation. I think an example will clarify this.
Given the recurrence equation
$$y[k+2]-4y[k]=\delta [k] + 2\delta [k+1]$$
with $$\delta [k]$$ being the discrete impulse, i.e $$\delta [k] = 1$$ for $$k=0$$ and $$\delta [k] = 0$$ for $$k\neq 0$$.
First you must find a solution for the homogeneous equation. I have no problems with that. But I don't know what substitution to use for finding the particular solution? Can someone help me with that?

Thanks,

Yoran

 

[jvicens]

Hello Yoran,

Thank you for your question. When dealing with recurrence equations with linear combinations of impulses on the right side, the most common approach is to use the method of undetermined coefficients. This method involves assuming a form for the particular solution and then solving for the coefficients.

In this particular case, we can assume a particular solution of the form y_p[k] = Ak\delta[k] + B(k+1)\delta[k+1], where A and B are constants to be determined. Plugging this into the original equation, we get:

A(k+2)\delta[k+2] + B(k+3)\delta[k+3] - 4(Ak\delta[k] + B(k+1)\delta[k+1]) = \delta[k] + 2\delta[k+1]

Simplifying and equating coefficients, we get:
A = 1/2 and B = -1/4

Therefore, the particular solution is y_p[k] = (1/2)k\delta[k] - (1/4)(k+1)\delta[k+1]

I hope this helps. Let me know if you have any further questions.