General Sir John Kotelawala Defence University
Faculty of Engineering
Department of Mathematics
Calculus – MA 1203
Tutorial 08
Year: 2023
Intake:40
Semester: 02
Stream: All
Learning outcomes covered: (LO5,LO6)
Name of the Instructor prepared: Ms. Erandi Karunawardana
Date: 19.10.2023
1. Find the Laplace transform of following functions.
i.
𝑓(𝑡) = 𝑆𝑖𝑛 2𝑡 𝐶𝑜𝑠 2𝑡
ii.
𝑓(𝑡) = 𝐶𝑜𝑠 2 3𝑡
iii.
𝑓(𝑡) = 𝑒 −𝑡 𝐶𝑜𝑠 2 𝑡
iv.
𝐶𝑜𝑠 𝜋𝑡, 𝑡 < 4
𝑓(𝑡) = {
0, 𝑡 ≥ 4
2. Evaluate
1
i.
𝐿−1 (𝑠(𝑠+1)(𝑠+2))
ii.
𝐿−1 ((𝑠−1)2 ( 𝑠+2))
iii.
𝐿−1 (𝑠3 −6𝑠2 +11𝑠−6)
4𝑠+5
2𝑠2 −6𝑠+5
3. Express the following function in terms of unit step function and find its Laplace transform.
𝑆𝑖𝑛 2𝑡,
i.
𝑓(𝑡) = {
0,
4
ii.
2𝜋 < 𝑡 < 4𝜋
𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
0 <𝑡 <1
𝑓(𝑡) = {−2
1<𝑡 <3
5
𝑡 >3
4. Solve the initial value problem
𝑦, =
i.
𝑥2
,
2𝑦 + 1
𝑦(0) = −1
Determine the general solution of the inhomogeneous linear differential equation
𝑦′ =
𝑥𝑦
1+ 𝑥 2
1+ 𝑥 2
+ √1− 𝑥2
by the method of integrating factor.
ii.
It is given that 𝑦 satisfies the differential equation
𝑑2 𝑦
𝑑𝑦
−5
+ 4𝑦 = 8𝑥 − 10 − 10 cos 2𝑥
2
𝑑𝑥
𝑑𝑥
a) Show that 𝑦 = 2𝑥 + Sin 2𝑥 is a Particular integral of given differential
equation.
b) Find the general solution of the differential equation.