1. Using the definition of the Laplace transform (LT) given in lectur...

1. Using the definition of the Laplace transform (LT) given in lectures, find the LT of the following functions: (a) f(t) = e -t cosh 2t. [3 marks] (b) g(t) = ( sin t cost, 0 = t = p/2, 0, t > p/2. [4 marks] Show all your working, including the evaluation of any integrals. Do not use tables of Laplace transforms. 2. Use partial fractions to find the inverse Laplace transform (ILT) of the following functions: (a) F(s) = s + 4 s 2 + 3s + 2 . [3 marks] (b) G(s) = 10s 2 - 24s + 9 s 2 (s - 3)2 . [3 marks] Where applicable, you may use entries (1) through to (10) and the basic general formulae from the table of Laplace transforms on MyUni. 3. Use Theorem 15 from the notes to find L -1 4 s 3 - 4s . [4 marks] Where applicable, you may use entries (1) through to (16) and the basic general formulae from the table of Laplace transforms on MyUni. Do not use partial fractions. Questions 4 and 5 on the next page. 4. Use Laplace transforms to solve the initial value problems (a) y 00 + 2y 0 + 17y = 34, y(0) = 0, y0 (0) = 0. [7 marks] (b) y 000 + 6y 00 + 12y 0 + 8y = 24te-2t , y(0) = 1, y0 (0) = -1, y00(0) = 2 [6 marks] Where applicable, you may use entries (1) through to (16) and the basic general formulae from the table of Laplace transforms on MyUni.

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School of Mathematical Sciences
Engineering Mathematics IIB, MATHS 2202
Assignment 1 question sheet
Due: 12:00 noon Tuesday 8 August (in the assignment drop boxes)
When presenting your solutions to the assignment, please include some explanation in
words to accompany your calculations. It is not necessary to write a lengthy description,
just a few sentences to link the steps in your calculation. Messy, illegible or inadequately
explained solutions may be penalised. The marks awarded for each part are indicated
in brackets.
1. Using the denition of the Laplace transform (LT) given in lectures, nd the LT
of the following functions:
t
(a) f(t) =e cosh 2t. [3 marks]
(
sint cost; 0t=2;
(b) g(t) = [4 marks]
0; t>=2:
Show all your working, including the evaluation of any integrals. Do not use
tables of Laplace transforms.
2. Use partial fractions to nd the inverse Laplace transform (ILT) of the following
functions:
s + 4
(a) F (s) = . [3 marks]
2
s + 3s + 2
2
10s 24s + 9
(b) G(s) = . [3 marks]
2 2
s (s 3)
Where applicable, you may use entries (1) through to (10) and the basic general
formulae from the table of Laplace transforms on MyUni.
3. Use Theorem 15 from the notes to nd
4
1
L :
3
s 4s
[4 marks]
Where applicable, you may use entries (1) through to (16) and the basic general
formulae from the table of Laplace transforms on MyUni.
Do not use partial fractions.
Questions 4 and 5 on the next page.4. Use Laplace transforms to solve the initial value problems
00 0 0
(a) y + 2y + 17y = 34; y(0) = 0; y (0) = 0: [7 marks]
000 00 0 2t 0 00
(b) y + 6y + 12y + 8y = 24te ; y(0) = 1; y (0) =1; y (0) = 2 [6 marks]
Where applicable, you may use entries (1) through to (16) and the basic general
formulae from the table of Laplace transforms on MyUni.
5. Use Laplace transforms and Theorem 15 from the notes to solve
Z
t
y(t) = 5 cos 2t 4 y() d
0
Where applicable, you may use entries (1) through to (16) and the basic general
formulae from the table of Laplace...