Understanding Engineering Mathematics

17.11 Reinforcement

1.Find the Laplace transforms of

(i) 2+t+ 3 t^2 (ii) 2 sin 3t+e−^2 t

(iii) etcos( 2 t) (iv) sin^2 t

2.Write down the inverse Laplace transforms of

(i)

1

s^4

+

8

s^3

(ii)

s^2 + 5 s+ 7

s^4

(iii)

1

s− 4

(iv)

1

s+ 4

(v)

2

4 s− 3

(vi)

c

as+b

(vii)

s

s^2 + 9

(viii)

6

s^2 + 9

(ix)

5 s+ 4

s^2 + 9

(x)

as+ 6

s^2 +c^2

wherea,b,care constants.

3.Find the inverses of the following Laplace transforms:

(i)

s+ 3

s(s+ 2 )

(ii)

1

(s+ 1 )(s− 3 )

(iii)

s^2 + 2

(s^2 + 1 )(s+ 1 )

(iv)

1

s^2 (s− 4 )

(v)

1

(s− 1 )(s+ 2 )^2

(vi)

1

(s^2 + 4 )(s^2 + 9 )

4.Solve the following initial value problems using the Laplace transform, and the results
of Question 3.

(i) y′+ 2 y= 3 y( 0 )= 1

(ii) y′− 3 y=e−t y( 0 )= 0

(iii) y′+y=2sinty( 0 )= 2

(iv) y′+ 4 y=ty( 0 )= 0

(v) y′−y=te−^2 t y( 0 )= 0

(vi) y′′+ 9 y=3sin2ty( 0 )=y′( 0 )= 0

In each case check your result by solution by another means (e.g. undetermined

coefficients).