Question

Solve un+2 + 10 un+1 + 9 un = 2^n.

(a) \(u_n = \frac{2^{n+1}}{33}+\frac{(-9)^{n+1}}{88}+\frac{(-1)^{n+1}}{24} \)

(b) \(u_n = \frac{2^n}{33}+\frac{(-9)^n}{88}+\frac{(-1)^{n-1}}{24} \)

(c) \(u_n = \frac{2^{n+1}}{11}+\frac{(-9)^{n+1}}{88}+\frac{(-1)^n}{24} \)

(d) \(u_n = \frac{2^n}{11}+\frac{(-9)^n}{88}+\frac{(-1)^{n-1}}{24} \)

This question was addressed to me in an online quiz.

This is a very interesting question from Linear Difference Equations and Z in section Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

Correct answer is (b) \(u_n = \frac{2^n}{33}+\frac{(-9)^n}{88}+\frac{(-1)^{n-1}}{24} \)

For explanation I would say: Take Z – Transformation on both sides. Now keep U(z) on one side and take everything else to other side. Now by partial fractions method, we get it in the format of a^n. Now, taking inverse Z-Transform on both sides, we get the required results.