Q10Advanced Engineering Mathematics

Difference equations are the discrete-time equivalent of differential equations, used extensively to model digital filters, population dynamics, and economic systems. The Z-transform is the premier mathematical tool for solving linear, constant-coefficient difference equations, perfectly analogous to how the Laplace transform is used for continuous differential equations.

1. Transforming the Difference Equation

We are presented with a second-order, non-homogeneous difference equation: $u_{n+2}-u_{n+1}+u_n=n^2{2}^n$. We are also provided with the critical initial conditions $u_0=0$ and $u_1=0$, which represent the state of the system at time zero.

The first step is to mathematically apply the one-sided Z-transform operator $Z\{\cdot \}$ to both sides of the equation. Let $Z\{u_n\}=U(z)$. We must rigorously apply the time-shifting properties of the Z-transform, which correctly incorporate the initial conditions:

• $Z\{u_{n+1}\}=zU(z)-z{u}_0$
• $Z\{u_{n+2}\}=z^{2}U(z)-z^2{u}_0-z{u}_1$

Because our initial conditions are all zero ($u_0=0,u_1=0$), these shifting formulas simplify remarkably to $Z\{u_{n+1}\}=zU(z)$ and $Z\{u_{n+2}\}=z^{2}U(z)$.

Next, we must find the Z-transform of the non-homogeneous forcing function on the right-hand side, $f(n)=n^2{2}^n$. We know the standard transform for $n^2$ is $Z\{n^2\}=\frac{z(z+1)}{(z-1{)}^3}$. We then apply the scaling property $Z\{a^{n}f(n)\}=F(z/a)$, with $a=2$.

Substituting all these transformed components back into the original difference equation yields an algebraic equation in the Z-domain:

2. Solving for U(z) and Inverse Transform

We algebraically factor out the unknown function $U(z)$ on the left side:

We then isolate $U(z)$ to obtain the system's transfer function multiplied by the input transform:

To find the time-domain solution $u_n$, we must perform the Inverse Z-Transform, $u_n=Z^{-1}\{U(z)\}$. The standard procedure is to divide $U(z)$ by $z$ and then perform a partial fraction expansion on the resulting complex rational function:

This expansion will involve finding the roots of the cubic term $(z-2{)}^3$ (yielding repeated roots at $z=2$) and the irreducible quadratic term $(z^2-z+1)$ (yielding complex conjugate roots $z=\frac{1\pm i\sqrt{3}}{2}$). The partial fraction decomposition takes the mathematical form:

After algebraically solving for the coefficients $A,B,C,D,E$ (which is a computationally intensive process of equating coefficients), we multiply the entire expression back by $z$. Finally, we apply standard inverse Z-transform formulas from lookup tables (e.g., $Z^{-1}\{\frac{z}{z-a}\}=a^n$) to each individual term to derive the final, exact closed-form expression for the discrete sequence $u_n$.