Question

What is the particular solution of the difference equation y(n)=\(\frac{5}{6}y(n-1)-\frac{1}{6}\)y(n-2)+x(n) when the forcing function x(n)=2^n, n≥0 and zero elsewhere?

(a) \(\frac{1}{5}\) 2^n

(b) \(\frac{5}{8}\) 2^n

(c) \(\frac{8}{5}\) 2^n

(d) \(\frac{5}{8}\) 2^-n

The question was posed to me by my school principal while I was bunking the class.

My doubt stems from Discrete Time Systems Described by Difference Equations in chapter Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (c) \(\frac{8}{5}\) 2^n

To explain I would say: The assumed solution of the difference equation to the forcing equation x(n), called the particular solution of the difference equation is

yp(n)=Kx(n)=K2^nu(n) (where K is a scale factor)

Upon substituting yp(n) into the difference equation, we obtain

K2^nu(n)=\(\frac{5}{6}\)K2^n-1u(n-1)-\(\frac{1}{6}\) K2^n-2u(n-2)+2^nu(n)

To determine K we must evaluate the above equation for any n>=2, so that no term vanishes.

=> 4K=\(\frac{5}{6}\)(2K)-\(\frac{1}{6}\) (K)+4

=> K=\(\frac{8}{5}\)

=> yp(n)=(8/5) 2^n.