- The document discusses different types of first order ordinary differential equations (ODEs) including separable, homogeneous, exact, and linear equations.
- It provides examples of identifying each type of equation and the general methods for solving them, such as using separation of variables, substitution to make equations homogeneous or separable, finding integrating factors, and determining if equations are exact.
- Various examples are worked through step-by-step to illustrate each problem solving technique. Exercises are also provided for students to practice applying the methods.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
This is a ppt of a college project of the topic kvl and kcl ..do read this..i have such interest in science projects and do maake aa lot of money by doing freelancing in embedded system so make sure to check this ppt for more updtes
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
We discuss the oscillatory behavior of all solutions of first order nonlinear neutral delay difference equations with variable coefficients of the form where are sequences of positive real numbers, and are sequences of nonnegative real numbers, and are positive integers. Our proved results extend and develop some of the well-known results in the literature. Examples are inserted to demonstrate the confirmation of our new results
This is a ppt of a college project of the topic kvl and kcl ..do read this..i have such interest in science projects and do maake aa lot of money by doing freelancing in embedded system so make sure to check this ppt for more updtes
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
We discuss the oscillatory behavior of all solutions of first order nonlinear neutral delay difference equations with variable coefficients of the form where are sequences of positive real numbers, and are sequences of nonnegative real numbers, and are positive integers. Our proved results extend and develop some of the well-known results in the literature. Examples are inserted to demonstrate the confirmation of our new results
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Introduction to AI for Nonprofits with Tapp Network
Scse 1793 Differential Equation lecture 1
1. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
1
- Ordinary Differential Equations (ODE)
Contains one or more dependent variables
with respect to one independent variable
- Partial Differential Equations (PDE)
involve one or more dependent variables
and two or more independent variables
is the dependent variable
while is the independent
variable
is the dependent variable
while is the independent
variable
Can you determine which one is the DEPENDENT VARIABLE and which
one is the INDEPENDENT VARIABLES from the following equations ???
Classification by type
Dependent Variable: w
Independent Variable: x, t
Dependent Variable: u
Independent Variable: x, y
Dependent Variable: u
Independent Variable: t
2. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
2
- Order of Differential Equation
Determined by the highest derivative
- Degree of Differential Equation
Exponent of the highest derivative
Examples:
a)
b)
c)
d)
- Linear Differential Equations
Dependent variables and their derivative are of degree 1
Each coefficient depends only on the independent variable
A DE is linear if it has the form
Examples:
1) 2)
3)
Order : 1 Degree: 2
Order : 2 Degree: 1
Order : 2 Degree: 1
Order : 3 Degree: 4
Classification by order / degree
Classification as linear / nonlinear
3. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
3
- Nonlinear Differential Equations
Dependent variables and their derivatives are not of degree 1
Examples:
1)
2)
3)
Initial conditions : will be given on specified given point
Boundary conditions : will be given on some points
Examples :
1) Initial condition
2) Boundary condition
Initial Value Problems (IVP)
Initial Conditions:
Boundary Value Problems (BVP)
Boundary Conditions:
Order : 1 Degree: 1
Order : 1 Degree: 2
Order : 3 Degree: 2
Initial & Boundary Value Problems
4. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
4
- General Solutions
Solution with arbitrary constant depending on the order
of the equation
- Particular Solutions
Solution that satisfies given boundary or initial conditions
Examples:
(1)
Show that the above equation is a solution of the following DE
(2)
Solutions:
(3)
(4)
Insert (1) and (4) into (2)
Proven that is the solution for the given DE.
EXERCISE:
Show that is the solution of the
following DE
Solution of a Differential Equation
5. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
5
Example 1:
Find the differential equation for
Solution:
( 1 )
( 2 )
Try to eliminate A by,
a) Divide (1) with :
( 3 )
b) (2)+(3) :
( 4 )
Example 2:
Form a suitable DE using
Solutions:
Forming a Differential Equation
6. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
6
Exercise:
Form a suitable Differential Equation using
Hints:
1. Since there are two constants in the general solution, has to
be differentiated twice.
2. Try to eliminate constant A and B.
7. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
7
1.2 First Order Ordinary Differential Equations (ODE)
Types of first order ODE
- Separable equation
- Homogenous equation
- Exact equation
- Linear equation
- Bernoulli Equation
1.2.1 Separable Equation
How to identify?
Suppose
Hence this become a SEPARABLE EQUATION if it can be written as
Method of Solution : integrate both sides of equation
8. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
8
Example 1:
Solve the initial value problem
Solution:
i) Separate the functions
ii) Integrate both sides
Answer:
iii) Use the initial condition given,
iv) Final answer
Note: Some DE may not appear separable initially
but through appropriate substitutions, the DE can be
separable.
Use your Calculus
knowledge to
solve this
problem !
9. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
9
Example 2:
Show that the DE can be reduced to a separable
equation by using substitution . Then obtain the solution
for the original DE.
Solutions:
i) Differentiate both sides of the substitution wrt
(1)
(2)
ii) Insert (2) and (1) into the DE
(3)
iii) Write (3) into separable form
iv) Integrate the separable equation
Final answer :
10. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
10
Exercise
1) Solve the following equations
a.
b.
c.
d.
2) Using substitution , convert
to a separable equation. Hence solve the original equation.
1.2.2 Homogenous Equation
How to identify?
Suppose , is homogenous if
for every real value of
Method of Solution :
i) Determine whether the equation homogenous or not
ii) Use substitution and in the original DE
iii) Separate the variable and
iv) Integrate both sides
v) Use initial condition (if given) to find the constant value
Separable
equation
method
11. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
11
Example 1:
Determine whether the DE is homogenous or not
a)
b)
Solutions:
a)
this differential equation is homogenous
b)
this differential equation is non-homogenous
12. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
12
Example 2:
Solve the homogenous equation
Solutions:
i) Rearrange the DE
(1)
ii) Test for homogeneity
this differential equation is homogenous
iii) Substitute and into (1)
iv) Solve the problem using the separable equation method
Final answer :
13. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
13
Example 3:
Find the solution for this non-homogenous equation
(1)
by using the following substitutions
(2),(3)
Solutions:
i) Differentiate (2) and (3)
and substitute them into (1),
ii) Test for homogeneity,
iii) Use the substitutions and
iv) Use the separable equation method to solve the
problem
Ans:
Note:
Non-homogenous can be reduced to a homogenous
equation by using the right substitution.
REMEMBER!
Now we use
instead of
LASTLY, do not forget to
replace with
14. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
14
1.2.3 Exact Equation
How to identify?
Suppose ,
Therefore the first order DE is given by
Condition for an exact equation.
Method of Solution (Method 1):
i) Write the DE in the form
And test for the exactness
ii) If the DE is exact, then
(1), (2)
To find , integrate (1) wrt to get
(3)
15. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
15
iii) To determine , differentiate (3) wrt to get
iv) Integrate to get
v) Replace into (3). If there is any initial conditions given,
substitute the condition into the solution.
vi) Write down the solution in the form
, where A is a constant
Method of Solution (Method 2):
i) Write the DE in the form
And test for the exactness
ii) If the DE is exact, then
(1), (2)
iii) To find from , integrate (1) wrt to get
(3)
iv) To find from , integrate (2) wrt to get
(4)
v) Compare and to get value for and .
vii) Replace into (3) OR into (4).
viii) If there are any initial conditions given, substitute the conditions into
the solution.
ix) Write down the solution in the form
, where A is a constant
16. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
16
Example 1:
Solve
Solution (Method 1):
i) Check the exactness
Since , this equation is exact.
ii)Find
(1)
(2)
To find , integrate either (1) or (2), let’s say we take (1)
(3)
iii) Now we differentiate (3) wrt to compare with
(4)
Now, let’s compare (4) with (2)
iv) Find
v) Now that we found , our new should looks like this
17. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
17
vi) Write the solution in the form
, where is a constant
Exercise :
Try to solve Example 1 by using Method 2
Answer:
i) Check the exactness
Since , this equation is exact.
ii) Find
(1)
(2)
To find , integrate both (1) and (2),
(3)
(4)
18. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
18
iii) Compare to determine the value of and
Hence,
and
iv) Replace into OR into (4)
v) Write the solution in the form
Example 2:
Show that the following equation is not exact. By using integrating
factor, , solve the equation.
Solution:
i) Show that it is not exact
Since , this equation is not exact.
Note:
Some non-exact equation can be turned into exact
equation by multiplying it with an integrating factor.
19. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
19
ii) Multiply into the DE to make the equation exact
iii) Check the exactness again
Since , this equation is exact.
iv) Find
(1)
(2)
To find , integrate either (1) or (2), let’s say we take (2)
(3)
v) Find
(4)
Now, let’s compare (4) with (1)
20. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
20
vi) Write
, where
Exercises :
1. Try solving Example 2 by using method 2.
2. Determine whether the following equation is exact. If it is,
then solve it.
a.
b.
c.
3. Given the differential equation
i. Show that the differential equation is exact. Hence, solve
the differential equation by the method of exact equation.
ii. Show that the differential equation is homogeneous.
Hence, solve the differential equation by the method of
homogeneous equation. Check the answer with 3i.
21. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
21
1.2.4 Linear First Order Differential Equation
How to identify?
The general form of the first order linear DE is given by
When the above equation is divided by ,
( 1 )
Where and
Method of Solution :
i) Determine the value of dan such the the coefficient
of is 1.
ii) Calculate the integrating factor,
iii) Write the equation in the form of
iv) The general solution is given by
NOTE:
Must be here!!
22. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
22
Example 1:
Solve this first order DE
Solution:
i) Determine and
ii) Find integrating factor,
iii) Write down the equation
iv) Final answer
Note:
Non-linear DE can be converted into linear DE by
using the right substitution.
23. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
23
Example 2:
Using , convert the following non-linear DE into linear DE.
Solve the linear equation.
Solutions:
i) Differentiate to get and replace into the non-
linear equation.
ii) Change the equation into the general form of linear
equation & determine and
iii) Find the integrating factor,
iv) Find
24. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
24
Since ,
v) Use the initial condition given, .
25. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
25
1.2.5 Equations of the form
How to identify?
When the DE is in the form
( 1 )
use substitution
to turn the DE into a separable equation
( 2 )
Method of Solution :
i) Differentiate ( 2 ) wrt ( to get )
( 3 )
ii) Replace ( 3 ) into ( 1 )
iii) Solve using the separable equation solution
26. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
26
Example 1:
Solve
Solution:
i) Write the equation as a function of
( 1 )
ii) Let and differentiate it to get
( 2 )
iii) Replace ( 2 ) into ( 1 )
iv) Solve using the separable equation solution
Substitution Method
27. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
27
Since ,
Since ,
28. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
28
1.2.6 Bernoulli Equation
How to identify?
The general form of the Bernoulli equation is given by
( 1 )
where
To reduce the equation to a linear equation, use substitution
( 2 )
Method of Solution :
iv) Divide ( 1 ) with
( 3 )
v) Differentiate ( 2 ) wrt ( to get )
( 4 )
vi) Replace ( 4 ) into ( 3 )
vii) Solve using the linear equation solution
Find integrating factor,
Solve
29. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
29
Example 1:
Solve
( 1 )
Solutions:
i) Determine
ii) Divide ( 1 ) with
( 2 )
iii) Using substitution,
( 3 )
iv) Replace ( 3 ) into ( 2 ) and write into linear equation form
v) Find the integrating factor
vi) Solve the problem
30. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
30
vii) Since
Exercise: Answer:
1.
2.
3.
4.
5.
6. Given the differential equation,
4 3 2 32 (4 ) 0.x y dy x y x dx
Show that the equation is exact. Hence solve it.
7. The equation in Question 6 can be rewritten as a Bernoulli equation,
12 4 .
dy
x y
ydx
By using the substitution 2
,z y solve this equation. Check the
answer with Question 6.
31. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
31
1.3 Applications of the First Order ODE
The Newton’s Law of Cooling is given by the following equation
Where
is a constant of proportionality
is the constant temperature of the surrounding medium
General Solution
Q1: Find the solution for
It is a separable equation. Therefore
Q2: Find when
Q3: Find . Given that
The Newton’s Law of Cooling
Do you know
what type of DE
is this?
32. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
32
Example 1:
According to Newton’s Law of Cooling, the rate of change of the temperature
satisfies
Where is the ambient temperature, is a constant and is time in minutes.
When object is placed in room with temperature C, it was found that the
temperature of the object drops from 9 C to C in 30 minutes. Then
determine the temperature of an object after 20 minutes.
Solution:
i) Determine all the information given.
Room temperature = C
When
When
Question: Temperature after 20 minutes,
ii) Find the solution for
iii) Use the conditions given to find and
When , C
When
33. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
33
iv)
Exercise:
1. A pitcher of buttermilk initially at 25⁰C is to be cooled
by setting it on the front porch, where the temperature
is . Suppose that the temperature of the buttermilk
has dropped to after 20 minutes. When will it be at
?
2. Just before midday the body of an apparent homicide
victim is found in a room that is kept at a constant
temperature of 70⁰F. At 12 noon, the temperature of the
body is 80⁰F and at 1pm it is 75⁰F. Assume that the
temperature of the body at the time of death was 98.6⁰F
and that it has cooled in accord with Newton’s Law.
What was the time of death?
34. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
34
The differential equation
Where
is a constant
is the size of population / number of dollars / amount of
radioactive
The problems:
1. Population Growth
2. Compound Interest
3. Radioactive Decay
4. Drug Elimination
Example 1:
A certain city had a population of 25000 in 1960 and a population of
30000 in 1970. Assume that its population will continue to grow
exponentially at a constant rate. What populations can its city
planners expect in the year 2000?
Solution:
1) Extract the information
2) Solve the DE
Separate the equation and integrate
Natural Growth and Decay
Do you know
what type of DE
is this?
35. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
35
3) Use the initial & boundary conditions
In the year 2000, the population size is expected to be
Exercise:
1) (Compounded Interest) Upon the birth of their first child, a couple
deposited RM5000 in an account that pays 8% interest compounded
continuously. The interest payments are allowed to accumulate. How
much will the account contain on the child’s eighteenth birthday? (ANS:
RM21103.48)
2) (Drug elimination) Suppose that sodium pentobarbitol is used to
anesthetize a dog. The dog is anesthetized when its bloodstream contains
at least 45mg of sodium pentobarbitol per kg of the dog’s body weight.
Suppose also that sodium pentobarbitol is eliminated exponentially from
the dog’s bloodstream, with a half-life of 5 hours. What single dose should
be administered in order to anesthetize a 50-kg dog for 1 hour? (ANS: 2585
mg)
3) (Half-life Radioactive Decay) A breeder reactor converts relatively stable
uranium 238 into the isotope plutonium 239. After 15 years, it is
determined that 0.043% of the initial amount of plutonium has
disintegrated. Find the half-life of this isotope if the rate of disintegration is
proportional to the amount remaining. (ANS: 24180 years)
36. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
36
Given that the DE for an RL-circuit is
Where
is the voltage source
is the inductance
is the resistance
CASE 1 : (constant)
(1)
i) Write in the linear equation form
ii) Find the integrating factor,
iii) Multiply the DE with the integrating factor
iv) Integrate the equation to find
Electric Circuits - RC
Do you know
what type of DE
is this?
37. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
37
CASE 2 : or
Consider , the DE can be written as
i) Write into the linear equation form and determine and
ii) Find integrating factor,
iii) Multiply the DE with the integrating factor
iv) Integrate the equation to find
(1)
Tabular Method
Differentiate Sign Integrate
+
+
-
38. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
38
(2)
From (1)
(3)
Replace (3) into (2)
Exercise:
1) A 30-volt electromotive force is applied to an LR series circuit in which
the inductance is 0.1 henry and the resistance is 15 ohms. Find the
curve if . Determine the current as .
2) An electromotive force
is applied to an LR series circuit in which the inductance is 20 henries
and the resistance is 2 ohms. Find the current if .
39. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
39
The Newton’s Second Law of Motion is given by
Where
is the external force
is the mass of the body
is the velocity of the body with the same direction with
is the time
Example 1:
A particle moves vertically under the force of gravity against air resistance ,
where is a constant. The velocity at any time is given by the differential
equation
If the particle starts off from rest, show that
Such that . Then find the velocity as the time approaches
infinity.
Solution:
i) Extract the information from the question
Initial Condition
ii) Separate the DE
Vertical Motion – Newton’s Second Law of Motion
40. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
40
Let ,
iii) Integrate the above equation
iv) Use the initial condition,
v) Rearrange the equation
Using Partial Fraction
41. CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793
41
vi) Find the velocity as the time approaches infinity.
When , =>