Some Key Concepts & Terms
Logarithm
Exponential
Derivative
Exponential and Logarithm
Primitive definitions:
( = 10 multiplied times, an integer )
Relations:
y
x== 1010101010
x
x
log
10 10
=
x
x
y
Exponential and Logarithm (2)
Mathematical definitions:
Relations:
N
N
x
N
x
e
+= → 1lim
xe
x
ln
=
Derivative
Rate of change wrt independent
variable; slope of curve:
Example:
( ) ( ) ( )
tytty
tdt
tdy
t−+
=→
1
lim 0
9
10
11
12
13
y
8
9
10
13
14
15
16
17
18
5
6
Examples of Derivatives
of Common Functions
Memorize these!
e
t
ty
kt
n
)(
1
ke
nt
dtdy
kt
n
−
____ _____
Integral
“Anti–derivative”; area under curve
where
( ) ( )
−
=
=
→ N
tt
tfduuf
N
i
iN
t
t
0
1
lim
0
10
11
12
13
10
12
13
14
16
17
Examples of integrals
of common functions
t
ty
n
)(
1
)1/(
)(
nt
dy
n
t
+
+
____ _____
Do not try to memorize these, but
ensure you know how to do them
Integration methods
For example, integration by parts:
Illustration:
=
udvdttt
t)(sin
0
Infinite Series
Taylor’s series examples
…
!
1
…
24
1
6
1
2
1
1
432
+++++++=
nt
t
n
tttte
Imaginary numbers
Example:
Solve
01
2
=++
xx
Classes of Differential Equations
Definitions and Examples
Differential Equation:
An equation involving derivatives.
Definitions and Examples (2)
Partial Differential Equation:
A differential equation involving two or more independent variables
Example: the beam equation
0
2
2
4
4
=
+
t
y
a
x
y
Definitions and Examples (3)
Order of a differential equation:
Order of the highest derivative in the equation
Examples:
First order
Second order
Definitions and Examples (4)
Linear differential equation (highly developed theory and solution methods):
General forms for linear ordinary differential equations (ODEs):
Nth order linear ODE:
Anything other is a nonlinear equation (complicated theory, often hard to
solve)
Definitions and Examples (5)
)(…
1
1
10 tgya
dt
yd
a
dt
yd
an
n
n
n
n
=+++ −
−
Linear Nth order ODEs are called time-invariant if
all the coefficients are constant:
Homework Assignment 1
Review the mathematical concepts discussed in today’s class
Read:
In the text, Chapter 1
Work: