The document is a problem sheet for a control systems analysis and design course. It contains 8 problems involving Laplace transforms and solving ordinary differential equations using Laplace transforms. The problems involve finding Laplace transforms and inverse Laplace transforms of various functions, using Laplace transforms to solve initial value problems for differential equations, and calculating the matrix exponential of given matrices.

1. Problem Sheet 1
DEN5200 Control Systems Analysis and Design
1. Find the Laplace transform of the following function:

 0
 (t < 1)

t+1 (1 ≤ t < 3)
f (t) =
 4
 (3 ≤ t ≤ 4)

0 (t > 4)
2. (a) Find the Laplace transform of f (t) = 1 + 7t2 + sin 2t + e−2t cos 3t.
s+2
(b) Find the inverse Laplace transform of F (s) = s2 +6s−7 .
s+1
(c) Find the inverse Laplace transform of F (s) = s2 −2s+5
.
3. Find the inverse Laplace transform of the following functions:
5 2s+3 10 2(s+1) 10
(a) (s+1)(s+2) (b) s3 +2s2 +s
(c) (s+2)(s2 +2s+5)
(d) s(s2 +s+2)
(e) s(s2 +4)(s+1)
4. Use Laplace transforms to solve the following ordinary differential equation
dy
− 2y = 3 cos t
dt
given y(0) = 2.
5. Use Laplace transforms to find the solution of the following differential equation
d2 y
+ 9y = sin 2t
dt2
given that y(0) = 1 and y ′ (0) = 0.
6. Find the solution of the following differential equation
d3 x d2 x
− 3 2 + 4x = 0
dt3 dt
2
given that x = 1, = −1 and d 2 = 0 at t = 0.
dx
dt dt
x
( )
At , where A = −5 3
7. Calculate e .
−2 0
( )
At , where A = 14 10
8. Calculate e .
−20 −14
1