ENGR201 2021 Q2 (ENGR5001)

ENGR201 2021 Q2Past paperOld spec ENGR2012:125 marks30 min

Consider the initial value problem of a dynamic system subjected to the exponential input $e^{2t}$ :

$\frac{d^2y}{dt^2} - 2\frac{dy}{dt} + y = e^{2t}$

with initial conditions $y(0) = \dfrac{dy(0)}{dt} = 0$ .

Formulas you may need

$\mathcal{L}[y'] = sY - y(0)$ , $\mathcal{L}[y''] = s^2Y - sy(0) - y'(0)$ (learn this; printed in the ENGR201 Laplace table)
$e^{at} \leftrightarrow \dfrac{1}{s - a}$ , $t\,e^{at} \leftrightarrow \dfrac{1}{(s - a)^2}$ (learn this; printed in the ENGR201 Laplace table)
Impulse $\delta(t - a) \leftrightarrow e^{-as}$ ; delay $H_a(t)g(t - a) \leftrightarrow e^{-as}G(s)$ (learn this; printed in the ENGR201 Laplace table)
Repeated-root partial fractions $\dfrac{A}{s - 2} + \dfrac{B}{s - 1} + \dfrac{C}{(s - 1)^2}$ (learn this)

(a)
Use the Laplace transform to solve the initial value problem (show your workings).
y(1) =-
[18]2:2
(b)
If the system is subjected to an additional impulse input $\delta(t - 2)$ , use the Laplace transform again to solve the problem (show your workings).
y(3) =-
[5]
(c)
Will the solution from (b) be discontinuous at $t = 2$ ? Explain your answer.
[2]