ChE 475

Process Dynamics and Control

Problem Set 3

October 20, 2014

1) Find the Laplace transform of

i) f(t)=cos(wt)

ii) f(t)=cosh(at)

iii) f(t)=5 cos(4t) + e-t + 5t

2) Invert the following Laplace transforms

i) F(s) =s + 2

s(s + 1)(s + 3) ii) F(s) =

1

(s + 2) (s + 3)3

iii) F(s) =10

(s + 6s + 2)(s + 2)2 iv) F(s) =

100

(s + 25)(s + 2)2

v) F(s) = (s + 3s + 4s + 5)e

s(s + 1)

3 2 -8s

3) Solve the following integro- differential equation using Laplace transformation.

dy

dty(t' ) dt' t

0

t

With y (0) =3

4) Find the complete time domain solutions for the following ODEs using Laplace

transforms.

i) texdt

xd 8

3

3

With x (0) = 0dt

)0(xd

dt

)0(dx2

2

ii) )2()2(3sin12 tUtxdt

dx With x (0) = 1

iii) t

2

2

ex25dt

dx6

dt

xd With x (0) = 0dt

)0(dx

5) For a system described by the following equations

t3eyxdt

dx x(0) = 1

0x4ydt

dy y(0) = 0

i) Find X(s) (be sure to eliminate Y(s) from this expression)

ii) Determine the final value of x(t) as t (both in time and s-domain)

iii) Find y (t).

6) Problems 1, 2, 3, 4, 5, 6 on page 267-268 of Chau