1-11
dx
=
dt
3 B -4 C
cos 3 t
-
4B+3C
sin 3 t
Solutions to Problems
-4t
e
.
Solving for the arbitrary constants, x(0) = 3B - 4C = 0. Therefore, B = -8/15. The final solution is
2
x (t) e
5
⎛ 8 sin(3t) 2
⎝
 15cos(3t⎞

5
 4t 
20.
⎠

a. Assume a particular solution of
Substitute into the differential equation and obtain
Equating like coefficients,
From which, C = -
1
5
1
.
and D = - 10
The characteristic polynomial is
Thus, the total solution is
1
= 2. Therefore, A =
5
Solving for the arbitrary constants, x(0) = A -
11
5
. Also, the derivative of the
solution is
dx
dt
.
Solving for the arbitrary constants, x(0) = - A + B - 0.2 = -3. Therefore, B =
3
. The final solution
5
is
x (t)
1
cos(2t)
1
5
⎛ 11cos(t)

3
sin(2t) e t 
⎝ 5sin(t⎞

5
10
b. Assume a particular solution of
xp = Ce-2t + Dt + E
Substitute into the differential equation and obtain
Copyright ©
2011 by John Wiley & Sons, Inc.
⎠

1-12
Chapter 1:
Introduction
Equating like coefficients, C = 5, D = 1, and 2D + E = 0.
From which, C = 5, D = 1, and E = - 2.
The characteristic polynomial is
Thus, the total solution is
Solving for the arbitrary constants, x(0) = A + 5 - 2 = 2 Therefore, A = -1. Also, the derivative of the
solution is
dx
dt
t 
t 
 ( A B)e Bte 10e 
 2t 
1 
. 
Solving for the arbitrary constants, x(0) = B - 8 = 1. Therefore, B = 9. The final solution is
c. Assume a particular solution of
xp = Ct 2 + Dt + E
Substitute into the differential equation and obtain
1
, D = 0, and 2C + 4E = 0.
4
1
1
, D = 0, and E = .
4
8
Equating like coefficients, C =
From which, C =
The characteristic polynomial is
Thus, the total solution is
Solving for the arbitrary constants, x(0) = A -
1
8 = 1 Therefore, A =
9
8. Also, the derivative of the
solution is
dx
dt
.
Solving for the arbitrary constants, x(0) = 2B = 2. Therefore, B = 1. The final solution is
Copyright ©
2011 by John Wiley & Sons, Inc.
1-13
Solutions to Problems
21.
Spring
displacement
Desired
force
Input
Input
transducer
Fup
voltage +
Controller
Pantograph
dynamics
Actuator
Fout
Spring
-
Sensor
22.
Desired
Amount of
HIV viruses
Amount of
HIV viruses
RTI
Patient
Controller
PI
Copyright ©
2011 by John Wiley & Sons, Inc.
1-14
Chapter 1:
Introduction
23.
a.
Desired
Climbing &
Rolling
Resistances
Controlled
Voltage
Inverter
Control
Command
Speed
ECU
Inverter
Electric
Motor
Motive
Force
Actual
Vehicle
+
+
Aerodynamic
Aerodynamic
Speed
Copyright ©
2011 by John Wiley & Sons, Inc.