Solve Difference Equation: y(n)-y(n-1)+0.25y(n-2)=x(n)-0.25x(n-1)

[sami_boy]

Hi guys

I have this difference equation.

y(n)-y(n-1)+0.25y(n-2)=x(n)-0.25x(n-1)

I found the step response of the system from the homogeneous solution and it's

y_h(n)=C₁(0.5)ⁿ+C₂n(0.5)ⁿ >>>homogeneous

h(n)=(0.5n+1)(0.5)ⁿ >>> step response

The 2nd part of the problem is I'm supposed to find the response of the system if x(n)=(0.25)ⁿu(n) is the input and initial conditions =0.

so my question is how do I find the particular solution ( y_p(n))?
I tried for y_p(n)=C(0.25)ⁿ,Cn²(0.25)ⁿ, but i couldn't find the response.
I also tried to solve it with convolution but it didn't work.

So please guys i would really appreciate your help.
thanks.

 

[jvicens]

Hi there,

Thank you for sharing your question with us. I am a scientist and I would be happy to help you with finding the particular solution for your difference equation.

First, let's define the variables in your equation for clarity:

y(n) - output at time n
x(n) - input at time n
u(n) - unit step function
C1, C2 - constants

To find the particular solution (yp(n)), we can use the method of undetermined coefficients. This method involves assuming a form for the particular solution and then solving for the coefficients based on the input and initial conditions.

In your case, the input is x(n) = (0.25)nu(n). Let's assume that the particular solution has the form yp(n) = Anu(n), where A is the coefficient we need to solve for. Substituting this form into the difference equation, we get:

Anu(n) - Anu(n-1) + 0.25Anu(n-2) = (0.25)nu(n) - 0.25(0.25)nu(n-1)

Simplifying this, we get:

An = 1/2n - 1/4(1/2)n-1

Now, we can use the initial conditions (y(0) = 0 and y(1) = 0) to solve for the constant A. Plugging in n=0 and n=1 into the above equation, we get:

A = 0
A = -1/4

These two equations are contradictory, so we can conclude that there is no particular solution for this equation. This means that the solution to the difference equation is the sum of the homogeneous solution (yh(n)) and the particular solution (yp(n)).

In summary, the solution to your difference equation is:

y(n) = yh(n) + yp(n) = C1(0.5)n + C2n(0.5)n

I hope this helps you solve your problem. If you have any further questions, please don't hesitate to ask.