Question

What is the homogenous solution of the system described by the first order difference equation y(n)+ay(n-1)=x(n)?

(a) c(a)^n(where ‘c’ is a constant)

(b) c(a)^-n

(c) c(-a)^n

(d) c(-a)^-n

The question was posed to me in an internship interview.

My question comes from Discrete Time Systems Described by Difference Equations in division Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

The correct choice is (c) c(-a)^n

To explain I would say: The assumed solution obtained by assigning x(n)=0 is

yh(n)=λ^n

=>y(n)+ay(n-1)=0

=>λ^n+a λ^n-1=0

=>λ^n-1(λ+a)=0

=>λ=-a

=>yh(n)=cλ^n=c(-a)^n