# 2021 Fall ME 6003 Applied Math 091321

```Lecture 01
Ordinary Differential Equations
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
10
How are the laws of nature formulated
mathematically?
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
11
Differential Equations
Physical laws of nature involve derivates
(rates). As such, they are formulated as
differential equations.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
12
Definition
A differential equation is an equation for a
missing function (or collection of functions)
in terms of the derivatives of those functions.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
13
The Modeling Process
The DE is part of the modelling process:
• Formulate a physical
mathematical terms
• The result is often a DE
• Solve the Equation
• Interpret the results.
Fall 2021
problem
Applied Mathematics
Lecture 01: Ordinary Differential Equations
in
14
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
15
Solving the differential equation means
finding an explicit function such that when it
substituted back into the DE, the equation
becomes an identity.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
16
Unlimited Population Growth
The rate at which a population of a species
grows at a certain time is proportional to the
total population at that time.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
17
Newton’s Law of Cooling
The rate at which a bodies temperature
changes is proportional to the difference
between the body temperature and the
temperature of the surrounding medium
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
18
Draining of a Tank
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
19
A Vibrating Mass
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
20
It is important to remember that the
differential equation is an expression of a
physical situation (at least, in engineering).
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
21
Some Calculus Reminders
Consider the function:
The derivative is:
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
22
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
23
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
24
The Chain Rule
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
25
Types of DEs
• Ordinary differential equation (ODE): Derivatives
are with respect to a single independent variable
• Partial differential equation (PDE): Derivatives are
with respect to two or more independent variables
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
26
Orders of DEs
The order of an ODE or PDE is the order of the
highest derivative in the equation
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
27
Forms
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
28
Linearity
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
29
Characteristics of Liner Eqns
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
30
Non-Linear Equations
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
31
Solution of an ODE
Any function , defined on an interval I and possessing at
least n derivatives that are continuous on I, which when
substituted into an nth-order ODE reduces the equation to
an identity
• Interval I can be an open interval (a, b), a closed
interval [a, b], an infinite interval (a, ), etc.
• A solution of a differential equation that is
identically zero on an interval I is a trivial solution
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
32
Solution Verification
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
33
The Solution Curve
The graph of a solution  of an ODE is a solution curve
and it is continuous on its interval I while the domain of
 may differ from the interval I
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
34
An explicit solution is one in which the dependent
variable is expressed solely in terms of the
independent variable and constants
• G(x,y) = 0 is an implicit solution if at least one
function  exists that satisfies the relation G and
the ODE on I.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
35
Families of Solutions
Similar to integration, we usually obtain a solution to a
first-order differential equation containing an arbitrary
constant c
A solution with a constant c represents a set of solutions,
called a one-parameter family of solutions
An n-parameter family of solutions solves an nth-order
differential equation
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
36
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
37
Initial Value Problems
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
38
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
39
Example 1.1
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
40
Existence &amp; Uniqueness
When solving an IVP, consider 2 fundamental questions:
• Does a solution exist?
• If a solution exists, is it unique?
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
41
Existence: Does the differential equation
possess solutions and do any of the solution
curves pass through the point (x0, y0)?
Uniqueness: When can we be certain there is
precisely one solution curve passing through
the point (x0, y0)?
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
42
From Ben-Dor, Shock Wave Reflection Phenomena
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
43
Example 1.2
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
44
A Theorem
Let R be a rectangular region in the x-y plane that
contains the point (x0,y0) in it’s interior. If
And
Are continuous on R, then there exists some
interval I and a unique function y(x) defined on I
that is a solution to the initial value problem.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
45
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
46
Visualizing Solution Curves
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
47
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
48
Consider
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
49
Example 1.3
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
50
Useful Websites
https://aeb019.hosted.uark.edu/dfield.html
https://www.monroecc.edu/faculty/paulseeburger/ca
lcnsf/DirectionField/
https://www.desmos.com/calculator/jabrxtyje0
https://homepages.bluffton.edu/~nesterd/apps/slopef
ields.html
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
51
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
52
Unconstrained Growth
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
53
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
54
Autonomous
Fall 2021
st
1
Order DEs
Applied Mathematics
Lecture 01: Ordinary Differential Equations
55
Limited Growth Model
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
56
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
57
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
58
Consider…
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
59
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
60
Solution Curves
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
61
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
62
Example 1.4
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
63
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
64
Sources and Sinks
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
65
Separable Equations
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
66
Example 1.5 - IVP
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
67
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
68
Example 1.6
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
69
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
70
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
71
Linear Equations
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
72
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
73
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
74
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
75
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
76
```