Chapter 08.01
Primer for Ordinary Differential Equations
After reading this chapter, you should be able to
:
1.
define an ordinary differential equation,
2.
differentiate between an ordinary and partial differential equation, and
3.
solve linear ordinary differential equations with fixed constants by using classical
solution and Laplace transform techniques.
Introduction
An equation that consists of derivatives is called a differential equation.
Differential
equations have applications in all areas of science and engineering.
Mathematical
formulation of most of the physical and engineering problems leads to differential equations.
So, it is important for engineers and scientists to know how to set up differential equations
and solve them.
Differential equations are of two types
(A)
ordinary differential equations (ODE)
(B)
partial differential equations (PDE)
An ordinary differential equation is that in which all the derivatives are with respect to a
single independent variable.
Examples of ordinary differential equations include
0
2
2
2
=
+
+
y
dx
dy
dx
y
d
,
4
)
0
(
,
2
)
0
(
=
=
y
dx
dy
,
,
sin
5
3
2
2
3
3
x
y
dx
dy
dx
y
d
dx
y
d
=
+
+
+
,
12
)
0
(
2
2
=
dx
y
d
2
)
0
(
=
dx
dy
,
4
)
0
(
=
y
Ordinary differential equations are classified in terms of order and degree.
Order
of an
ordinary differential equation is the same as the highest derivative and the
degree
of an
ordinary differential equation is the power of highest derivative.
Thus the differential equation,
x
e
xy
dx
dy
x
dx
y
d
x
dx
y
d
x
=
+
+
+
2
2
2
3
3
3
08.01.1
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Chapter 08.01
is of order 3 and degree 1, whereas the differential equation
x
dx
dy
x
dx
dy
sin
1
2
2
=
+
+
is of order 1 and degree 2.
An engineer’s approach to differential equations is different from a mathematician.
While,
the latter is interested in the mathematical solution, an engineer should be able to interpret the
result physically.
So, an engineer’s approach can be divided into three phases:
a)
formulation of a differential equation from a given physical situation,
b)
solving the differential equation and evaluating the constants, using given conditions,
and
c)
interpreting the results physically for implementation.
Formulation of differential equations
As discussed above, the formulation of a differential equation is based on a given physical
situation.
This can be illustrated by a springmassdamper system.
Above is the schematic diagram of a springmassdamper system. A block is suspended
freely using a spring.
As most physical systems involve some kind of damping  viscous
damping, dry damping, magnetic damping, etc., a damper or dashpot is attached to account
for viscous damping.
Let the mass of the block be
M
, the spring constant be
K
, and the damper
coefficient be
b
.
If we measure displacement from the static equilibrium position we need
not consider gravitational force as it is balanced by tension in the spring at equilibrium.
Below is the free body diagram of the block at static and dynamic equilibrium.
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 Spring '07
 Gallivan
 Laplace, Sin, ........., Partial differential equation

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