QUESTION $2$

Evaluate the following equations using Laplace transform:

$($a$)$

$f^{\text{'}\text{'}}(x)+5f^{\text{'}}(x)+6f(x)=2e^{-x}$ where $f(0)=0$ and $f^{\text{'}}(0)=0$

$($b$)$

$f^{\text{'}\text{'}}(x)-4f(x)=sin2x$ where $f(0)=1$ and $f^{\text{'}}(0)=-2$

Find the inverse transform of:

$Y(s)=\frac{1+e^{-2s}}{(4s+1)(3s+1)}$

$[15$ Marks$]$

QUESTION $3$

Given the transfer function below, write it in standard first order form and determine

the time constant and the steady state gain:

$\frac{Y(s)}{x(s)}=\frac{2}{s+\frac{1}{3}}$

The transfer function that corresponds to the inlet volumetric flowrate $F_{in}(t)$ and the

outlet volumetric flowrate $F_{\text{o}\text{u}\text{t}}(t)$ of two mixing tanks in parallel is given below:

$G(s)=\frac{[(1-a)A_1{R}_1+a{A}_2{R}_2]s+1}{(A_1{R}_{1}s)(A_2{R}_{2}s+1)}$

What are the poles and zeros of the system? Are there any common factors? And under what

conditions?

$[16$ Marks$]$