Foundations (5)
In first order ordinary differential equations, we are answering questions
of the form:
If is a given function of time and of the integral , what is the
function ? That is, we are solving problems of the form
dtdy
y
)(ty
Some Key Concepts & Terms
Logarithm
Exponential
Exponential and Logarithm
Primitive definitions:
Relations:
y
x== 1010101010
x
x
log
10 10
=
Exponential and Logarithm (2)
Mathematical definitions:
Relations:
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
0
11
12
14
3
5
8
Examples of Derivatives
of Common Functions
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
13
15
17
Examples of integrals
of common functions
e
t
ty
kt
n
)(
1
/
)1/(
)(
ke
nt
dy
kt
n
t
+
+
____ _____
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:
Definitions and Examples (2)
Partial Differential Equation:
A differential equation involving two or more independent variables
Example: the beam equation
Definitions and Examples (3)
Order of a differential equation:
Order of the highest derivative in the equation
Examples:
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)
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:
Always read over the day’s lecture notes and be sure you understand them.