Q5Advanced Engineering Mathematics I

Question

Solve the following difference equation by using Z-transform: $u_{m + 2} - u_{m + 1} + u_{n} = n^{2} 2^{n}, u_{0} = u_{1} = 0$

Answer 1

Take the Z-transform of $u_{n + 2} - u_{n + 1} + u_{n} = n^{2} 2^{n}$ .

Substitute the initial conditions $u_{0} = 0, u_{1} = 0$ .

The RHS requires finding the Z-transform of $n^{2} 2^{n}$ , which involves applying the scaling property $Z {a^{n} x_{n}} = X (z / a)$ to the transform of $n^{2}$ .

Solve algebraically for $U (z)$ and invert using partial fractions.

Q5Advanced Engineering Mathematics I

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