APPLICATIONS
Law of Natural Growth or Decay
Let denote a certain quantity that depends on time,
denoted by the variable . Some examples are population in
a certain place, number of bacteria in a culture, amount of
radioactive material, number of items mem
Special Series
Convergence of Power Series
MATH115 ADVANCED ENGINEERING MATHEMATICS
Learning Objectives
At the end of the lecture presentation, the students
should be able to:
1. Recognize the three special kinds of series
(arithmetic, harmonic and geo
MATH115
Power Series Representation of Functions
Using power series representation for known functions
one can derive power series representation for other
functions by the following methods:
Substitution
Differentiation
Integration
Within the interv
EXACT
Differential Equation
Linear and First Order
Let us consider the family of functions . An example is .
Such a family of functions can be the general solution of a
first order linear ODE. For instance is the general solution
of the ODE
To prove this,
APPLICATIONS
Mixture Problems
Electrical Circuits
Mixture Problems
A container initially contains
gallons of brine with pounds of
salt per gallon of brine. A brine
solution with lb/gal of brine is
pumped into the container at a
constant rate of gal/min. T
APPLICATIONS
Newtons Law of Cooling/Heating
According to Newton, The rate of cooling or heating of an
object is proportional to the temperature difference
between the surrounding and the object.
If is the temperature of the object at time from the start
o
Differential
Equations
Ordinary and Partial
What is a differential equation?
The following are all differential equations:
Based on the examples given in the last slide, can you
define or explain, in your own words, the meaning of
differential equation?
A
Homogeneous
Differential Equation
Linear and First Order
Homogeneous Function
A function is homogeneous of degree if .
Examples.
1. is homogeneous of degree 2 because
2. is homogeneous of degree because
3. is homogeneous of degree 3.
4. is not homogeneous
First Order
Differential Equation
Separation of Variables
Consider the following first order differential equations:
and
In (a) we can integrate readily because the expression on
the right is a function of only. That is .
In (b) we cannot write . (Explain
APPLICATIONS
Dynamics/Escape Velocity
Dynamics
One famous formula in dynamics due to Newton is .
Example. (Problem 17, page 67) In the motion of an object
through a certain medium (air at certain pressures for
example), the medium furnishes a resisting fo
Bernoullis
Equation
The Bernoulli differential equation is the equation
Where is a positive integer. Note that when , the
differential equation is linear and can be solved by
separation of variables. Can you verify this statement?
Let us consider the case
ODE Solvable by
Integrating Factor
Linear and First Order
The general linear ordinary differential equation of order 1
has the form
If we divide by , we get the form
Where and .
Let and multiply the differential equation by The result is
an exact differen