Higher Engineering Mathematics

Assign-18-H8152.tex 23/6/2006 15: 17 Page 655

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Laplace transforms

Assignment 18

This assignment covers the material contained

in Chapters 64 to 68.

The marks for each question are shown in

brackets at the end of each question.

1. Find the Laplace transforms of the following
   functions:
   (a) 2t^3 − 4 t+ 5 (b) 3e−^2 t−4 sin 2t
   (c) 3 cosh 2t (d) 2t^4 e−^3 t
   (e) 5e^2 tcos 3t (f) 2e^3 tsinh 4t (16)


2. Find the inverse Laplace transforms of the fol-
   lowing functions:

(a)

5

2 s+ 1

(b)

12

s^5

(c)

4 s

s^2 + 9

(d)

5

s^2 − 9

(e)

3

(s+2)^4

(f)

s− 4

s^2 − 8 s− 20

(g)

8

s^2 − 4 s+ 3

(17)

3. Use partial fractions to determine the following:

(a) L−^1

{

5 s− 1

s^2 −s− 2

}

(b) L−^1

{

2 s^2 + 11 s− 9

s(s−1)(s+3)

}

(c) L−^1

{

13 −s^2

s(s^2 + 4 s+13)

}

(24)

4. In a galvanometer the deflectionθsatisfies the
   differential equation:

d^2 θ

dt^2

+ 2

dθ

dt

+θ= 4

Use Laplace transforms to solve the equation for

θgiven that whent=0,θ= 0 and

dθ

dt

= 0

(13)

5. Solve the following pair of simultaneous differ-
   ential equations:

3

dx

dt

= 3 x+ 2 y

2

dy

dt

+ 3 x= 6 y

given that whent=0,x=1 andy=3. (20)

6. Determine the poles and zeros for the trans-

fer function:F(s)=

(s+2)(s−3)

(s+3)(s^2 + 2 s+5)

and plot

them on a pole-zero diagram. (10)