Q3Advanced Engineering Mathematics I

Question

Solve by z transform of $u_{n + 2} - 6 u_{n + 1} + 8 u_{n} = 2^{n} + 6 n$ .

Answer 1

Equation: $u_{n + 2} - 6 u_{n + 1} + 8 u_{n} = 2^{n} + 6 n$ .

Taking the Z-transform of both sides: $z^{2} [U (z) - u_{0} - u_{1} z^{- 1}] - 6 z [U (z) - u_{0}] + 8 U (z) = Z {2^{n}} + 6 Z {n}$

Where $Z {2^{n}} = \frac{z}{z - 2}$ and $Z {n} = \frac{z}{( z - 1 ) ^{2}}$ .

After substituting initial conditions (if provided, else they remain constants $C_{1}, C_{2}$ ), we solve for $U (z)$ and find the inverse Z-transform using partial fraction decomposition.

Q3Advanced Engineering Mathematics I

Question

Answer