1) The document presents information on ordinary differential equations including definitions, types, order, degree, and solution methods.
2) Differential equations can be written in derivative, differential, and differential operator forms. Common solution methods covered are variable separable, homogeneous, linear, and exact differential equations.
3) Applications of differential equations include physics, astronomy, meteorology, chemistry, biology, ecology, and economics for modeling various real-world systems.
Differential equation and its order and degreeMdRiyad5
The order of a differential equation is determined by the highest-order derivative; the degree is determined by the highest power on a variable. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution
Differential equation and its order and degreeMdRiyad5
The order of a differential equation is determined by the highest-order derivative; the degree is determined by the highest power on a variable. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution
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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. Ordinary differential equation
Differential equation:
It contains derivatives of unknown or
An equation which involves differential Co-efficient is called a differential
equation.
Independent variable is always in the bottom and dependent variable on the
top.
2
2
d y dy
0
dxdx
Ex:
x is the independent variable
y is the dependent variable
2
2
dy 1 x dy
, f x,y
dx dx1 y
3. Differential equations can be written as three different forms
1. Derivative form:
In derivative form the equation look likes
y'' y' 0
2.Differential form
In differential form the equation look likes
2
2
d y dy
0
dxdx
3.Differential operator
In differential operator form the equation look likes
2
D y Dy 0
4. Application:
1. Exponential Growth – Population
Let P(t) be a quantity that increases with time t and the rate of increase is proportional to the same
quantity P as follows
Where is the first derivative of P, k > 0 and
t is the time.
The solution to the above first order differential equation is given by
Where A is a constant not equal to 0.
If P = P0 at t = 0, then
dp
kp
dt
dp
dt
k t
P t A e
0
0P A e
5. which gives A = P0
The final form of the solution is given by
k t
0P t P e
Assuming P0 is positive and since k is positive, P(t) is an increasing exponential.
dp
kp
dt
is also called an exponential growth model.
2. Determine the velocity of a moving object.
We are going to use the example of a car travelling along a road. Where x-axis
represent the position of the car.
6. Change in position
Change in time
dx 4
dt 5
dx 4
dt 5
dx 4mile
5mindt
multiply by 12 we get
dx 48mile
60mindt
Velocity of the car
dx
48mph
dt
Velocity is the first derivative of position of x and it is represented by dx
dt
7. Ordinary Differential equation (ODE)
ODE is an equation containing a equation of one independent variable & its
derivatives
OR
ODE is an equation in which unknown equation depends on single independent
variables
OR
If is on unknown equation, an equation which involves at least one
derivative of y. w.r.t x. is called ODE.
Ex:
y y x
y f x
2
2
d y 2dy
8y 0
dxdx
8. Partial differential equation (PDE)
PDE is a differential equation in which known equation depends on several
independent variables
be a equation of two independent variable x and y.
Then, partial derivative of z with respect to x keeping y constant is
Similarly z w.r.t y keeping x constant is
Definition : An equation involving partial derivatives is called a partial differential
equation (PDE)
Let z f x,y
z
x
z
y
2 2 2
2 2
z z z z z
p ,q ,r ,s ,t
x y x yx y
Ex:
z z z
1) a 2) z xy
x x y
9. Order:
The orders of the differential equation is the order of the highest derivative present
in the equation.
Degree:
The degree of the differential equation is the degree of the highest order derivative
after clearing the fractional powers or removing radical signs.
Ex:
1. [order =1, degree=2]
2. [order=3, degree=1]
3. [order=2, degree=1]
2
dy dy
3 2 0
dx dx
33 2
2 2
d y d y dy
5 sinx
dxdx dx
2
2
2
d y
w y 0
dx
10. Solutions of differential equation
General solution:
General solution is a solution is which the number of arbitrary constants is equal to
the order of the differential equation.
3x
y eEx: is solution of D.E
3x
3x
3x
3x 3x 3x 3x
y e 0
dy
3e 0
dx
dy
3e 0
dx
d
e 3 e 3e 3e 0
dx
11. 3xdy
3e
dx
Integrate both sides we get
3x
3x
3e
y c
3
y e c
Particular solution:
Particular solution is a solution obtained from general solution by given particular
value to arbitrary constants.
Initial condition 3x
y 0 1 y e
y 1 x 0when
3x
0
3x
y e c
1 e c
1 1 c 0 c
y e 0
12. Singular solution:
It does not contain any arbitrary constants.
Solution of the D.E is
dy a
y x
dydx
dx
2
y 4ax which does not contain any arbitrary constant.
Both ODE & PDE equation are broadly classified as linear & non
liner differential equation.
Linear differential equation
A differential equation is linear if the unknown equation and its derivate appear
to the power 1.
or
13. A differential equation is linear, if it can be expressed as
n n 1 n 2
0 1 2 nn n 1 n 2
d y d y d y
a a a .......... a y b
dx dx dx
where 0 1 2 n..a ,a ,a ..a & b are constants or function of x. clearly this differential
equation is nth order and degree one.
Ex:
2
2
d y dy
x y 0
dxdx
dy
x y 1
dx
Nonlinear differential equation
A Differential equation which is not linear is called a non-linear differential
equation.
Ex:
23
2
3
d y dy
y x
dxdx
14. Differential equation of the first orders & first degree
General O.D.E of nth order is
2 n
2 n
dy d y d y
F x,y, , , 0
dx dx dx
n
F x, y, y', y'' y 0
y' f x,y
Implicit Form
Explicit Form
dy
f x,y
dx
Or M x,y dx N x,y dy 0
16. Solution of differential equations of first order & first degree
1. Variables separable method
P x dx Q y dy 0 f x dx y dy
P x dy Q y dy c
Ex: 2 2
sec xtanydx sex ytanxdy 0
2. Homogenous Differential equation
A differential equation of the form is Called a homogenous equation, if
each term of and are homogenous equation of the same degree.
Or
A function is said to be a homogenous function of degree ‘n’ if
or
f x,ydy
dx x,y
f x,y x,y
U f x,y n y
U x g
x
n x
U y g
y
17. Ex: U 2x 3y
y y
U x 2 3 x'g
x x
U is a homogeneous function of degree 1
2 2
dy 2xy
dx x y
Ex:
We put y vx
We take the substitution & equation reduced to variable
separable method
dy xdx
v
dx dx
18. 3. Liner differential equation
A differential equation of the form dy
py
dx
Where P & Q are function of x or constants
pdx
e Integrating factor (I.F)
pdx pdx
Y IF Q I.F dx c
Y e Q e dx c
19. 4. Exact differential equation
An differential equation of the form is said to be on exact
differential equation, If it satisfies the condition
Mdx Ndy 0
M N
y x
The solution of above differential equation is OrMdx Ndy c Mdx N y dy c
y Constant terms of N free form X
Ex:
dy 2x y 1
dx x 2y 1
M 2x y 1 N x 2y 1
M N
1 and 1
y x
2 2
M N
y x
2x y 1 2y 1 dy c
x xy x y y c
20. Non-homogenous equation:
An equation of the form
dy ax by c
bdx a 'x ' c'
y
(Where are constant) is called a non-homogenous differential
equation of first order. To solve this type of equation.
a,b,c,a',b',c'
(1) If ax by k a'x b'y , The substitutionax by v so that
dy dv
a b
dx dx
In this case the equation reduce to variable separable form and hence it can be solved.
dy 1 dv
a
dx b dx
and
(2) If by cross multiplication and suitable resolving of the taxes and then
by doing term by term integration the equation can be solved.
"b a"
21. Solve: x y dx x y dy dx dy
Solution: x y 1 dx x y 1 dy
dy x y 1
dx x y 1
………….. (1)
x y v dy dv dy dv
1 1
dx dx dx dx
Put so that
Substituting in (1), we get dv v 1
1
dx v 1
dv v 1
1
dx v 1
dv 2v v 1
dv 2dx
dx v 1 v
1
1 dv 2 1dx v log v 2x c
v
Integrating
x y log x y 2x c General solution is