laplac

please see attached.

General Information

All questions in the tasks must be completed correctly with sufficient detail to gain the
pass criteria.

All submissions to be electronic in MS Word format with a minimum of 20 typed
words. Also add footer to the document with your name. All answers must be clearly
identified as to which task and question they refer to. All work must be submitted
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Task 1 – Learning Outcome 2.1

Determine Laplace transforms and their inverse using tables and partial
fractions

1. Find the Laplace transform of each of the following

a. 8 + 3𝑡𝑡 − 2𝑡𝑡
2

3
− 2𝑡𝑡6 + 𝑒𝑒3𝑡𝑡 + 𝑠𝑠𝑠𝑠𝑠𝑠3𝑡𝑡 − 𝑐𝑐𝑐𝑐𝑠𝑠ℎ2𝑡𝑡

b. �
3 𝑐𝑐𝑐𝑐𝑠𝑠2𝑡𝑡+2 𝑒𝑒3𝑡𝑡 +2 𝑡𝑡4 + 3𝑡𝑡+4

𝑒𝑒5𝑡𝑡
�

2. Find the inverse Laplace transforms of the following functions

a.

3
𝑠𝑠−2

+
2
𝑠𝑠+1

+
3

(𝑠𝑠+1)5

b.
2
𝑠𝑠

+
4
𝑠𝑠5

+
𝑠𝑠

𝑠𝑠2 +16

c. 𝑠𝑠+4
𝑠𝑠2 +4𝑠𝑠+4

d. 4
(𝑠𝑠−1)(𝑠𝑠+2)

Task 2 – Learning Outcome 2.2

Solve first and second order differential equations using Laplace transforms

3. Solve the following first order differential equations using Laplace transforms

𝑑𝑑𝑠𝑠
𝑑𝑑𝑡𝑡

+ 3𝑠𝑠 = 2𝑒𝑒−5𝑡𝑡 𝑠𝑠(0) = 3

4. Use Laplace transform to solve the following second order differential equation

𝑣𝑣′′ + 2𝑣𝑣′ + 𝑣𝑣 = 𝑡𝑡2 − 1 𝑣𝑣(0) = 1, 𝑣𝑣′ (0) = −1

Task 3 – Learning Outcome 2.3

Model and analyse engineering systems and determine system behaviour
using Laplace transforms

5. A series LR circuit with a step input voltage can be modelled by the equation:

𝐿𝐿

𝑅𝑅
𝑑𝑑𝑠𝑠
𝑑𝑑𝑡𝑡

+ 𝑠𝑠 = 𝑉𝑉
𝑅𝑅
, i(0)=0 L, R and V are constants

Use Laplace transforms to show:

𝑠𝑠 = 𝑉𝑉
𝑅𝑅
�1 − 𝑒𝑒−

𝑅𝑅
𝐿𝐿
𝑡𝑡�

6. The charge in a series LCR circuit is modeled by the equation:

𝐿𝐿 𝑑𝑑
2𝑞𝑞
𝑑𝑑𝑡𝑡2

+ 𝑅𝑅 𝑑𝑑𝑞𝑞
𝑑𝑑𝑡𝑡

+ 𝑞𝑞
𝐶𝐶

= 𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡 𝑞𝑞(0) = 0, 𝑞𝑞′(0) = 0

Solve the equation in case when L=2 H, R=20 Ω, C=0.02 F

End of assessment brief

General Information
Task 1 – Learning Outcome 2.1
Task 2 – Learning Outcome 2.2
,𝑑𝑖-𝑑𝑡.+3𝑖=2,𝑒-−5𝑡. 𝑖,0.=3
Task 3 – Learning Outcome 2.3