2010-13.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	13	IB 2010 Complex Methods	Paper 4, Section II, A	2010

A linear system is described by the differential equation

$$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=f(t),$$

with initial conditions

$$y(0)=0, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=1$$

The Laplace transform of $f(t)$ is defined as

$$\mathcal{L}[f(t)]=\tilde{f}(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

You may assume the following Laplace transforms,

$$\begin{aligned} \mathcal{L}[y(t)] &=\tilde{y}(s) \\ \mathcal{L}\left[y^{\prime}(t)\right] &=s \tilde{y}(s)-y(0) \\ \mathcal{L}\left[y^{\prime \prime}(t)\right] &=s^{2} \tilde{y}(s)-s y(0)-y^{\prime}(0) \\ \mathcal{L}\left[y^{\prime \prime \prime}(t)\right] &=s^{3} \tilde{y}(s)-s^{2} y(0)-s y^{\prime}(0)-y^{\prime \prime}(0) \end{aligned}$$

(a) Use Laplace transforms to determine the response, $y_{1}(t)$, of the system to the signal

$$f(t)=-2$$

(b) Determine the response, $y_{2}(t)$, given that its Laplace transform is

$$\tilde{y}_{2}(s)=\frac{1}{s^{2}(s-1)^{2}} .$$

(c) Given that

$$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=g(t)$$

leads to the response with Laplace transform

$$\tilde{y}(s)=\frac{1}{s^{2}(s-1)^{2}},$$

determine $g(t)$.