x (0) + x (0) A sin( t + ) xxx ++ 634 = 5sin6 t x (0)0 = x += 35 xt x (0)10 = et sin et cos

1. Derive the Laplace transforms of the exponentially decaying sine and cosine functions,

at

sin

e

⁻

ω t

^and

e

^−at

cos ω t

, where a and

ω

are constants.

cos t ω t

. 2. Derive the Laplace transform of the function

3. Determine the complete response of the following model, which has a ramp input:

x 3 x 5 t

•

+ =

x (0) 10 =

4. Use two methods to obtain the inverse Laplace transform of

2 2

3 7 3 7

24 136 4( 6 34)

s s

s s s s

( ) +

X s = 4 = +

+ + + +

5sin 6

5. Use the Laplace transform to solve the following problem:

6 34

x x x = t

+

•

+ x (0) 0

••

x (0) 0

=

•

=

6. (a) Derive the Laplace transform of the function

A sin( ω φ t + )

.

(b) Generalize the answer from part (a) to find the Laplace transform of

sin( ) Ae

⁻at

ω φ t +

.

7. Obtain the inverse Laplace transform of

1 2

( ) ( 4)

e s

X s s s

− −

= +

8. Suppose a rectangular pulse P(t) of unit height and duration 2 is applied to the first-order model

x 4 x P t ( )

•

+ =

with a zero initial condition. Use the Laplace transform to determine the response. (***Draw a graph of the response.***)

9. Obtain the transfer functions X(s)/F(s) and Y(s)/F(s) for the following model:

3 x y

•

= y f t ( ) 3 y 15 x

•

= − −

(0 ) x

(0) x

.

+

and 10. Solve the following Problems for x(t). Compare the values of

For parts (b) through (d), also compare the values of

x (0

•

x

) (0)

• .

+

and

a.

7 x

^•

+ 5 x = 4 ( ) δ t

x (0) = 3

b. ^••

x + 14 x

^•

+ 49 x = 3 ( ) δ t

x (0) = 2

x (0) 3

•

=

11. Obtain the Laplace transform of the function plotted in Figure 1.

12. A trapezoidal profile f(t) is shown in Figure 2. Obtain the response x(t) of the following model:

2 x x f ( )

•

+ = t

x (0) = 0

Fig 1. Fig 2.