laplac

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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